I'm revising my answer again, but this time with an opinion based on my observations of peppers (not bell types specifically):
My opinion is that they'll continue to live as long as they're healthy and pruned. If they're in a small container and not pruned, they might die after they produce ripe fruit the first or second time, but if you prune them afterward and keep them healthy, you should be able to get the plant to live for a very long time, even in a small container.
A general rule I go by is that if the leaves are starting to fall off, or if the plant isn't growing anymore, then prune it. (Provided it's otherwise healthy.) It should grow new leaves/branches, and flowers, too.
Picking the fruit when they're still green may encourage more fruit to develop in the long run. The plants may be more prone to dying directly after the fruit is all ripe than beforehand. Interrupting this process seems to be healthy for the plant (although if you're taking good care of the plant, this may not be much of a concern).
Peppers use nutrients fast, though, and they have a lot of pests. So, keeping them healthy can be a challenge, since they require regular maintenance. Knowing what nutrients to give them can also be a challenge. I've personally found that indoor peppers seem to improve with the addition of a little wood ash and other stuff.
Bell types will need larger containers, maybe more light, and probably different care, but I imagine the principles are similar.
I'm revising my answer significantly (my old answer is at the bottom, after the divide).
I believe you're looking for bell peppers that 'overwinter' well (probably more than once). Overwintering is usually when you pot your pepper and take it inside for the winter (or just keep it in a pot it's already in for the winter).
Originally, I thought that indeterminate peppers were peppers that would continue to grow and produce fruit indefinitely, and that any indeterminate sweet pepper would suffice in answering your question. However, some people have different definitions of what indeterminate means, apparently. Some say it just means that the plants produce flowers all throughout a single season, and then die like other annuals regardless of the climate (this largely comes from my contacting a Canadian website that spoke of greenhouse and field peppers, and their response to me, as well as Seed Savers Exchange—those who responded may not have had a great knowledge on the topic of the article, though, but seemed to have some good general knowledge about plants—but it's also a general observation that not everyone thinks it has anything to do with longevity or size, but rather bloom frequency only).
Some people say that all sweet peppers are indeterminate. Some say field peppers are determinate.
Some say all peppers are perennials, provided the climate is right. Some say all peppers are annuals (of course, they're wrong, but they might be right in that some peppers are annuals).
Here's what I've concluded (which may or may not be entirely accurate):
Most peppers are capable of surviving longer than one season (at least up until December), whether or not they do very well afterward. Some of them will survive up to 5+ years. One of the comments to your question, and another source (15 years of fruit; 10+ feet tall), speak of 10–15 year old peppers, but I haven’t specifically heard of anyone growing one in-doors for that long, yet (although I personally believe it can happen). Some varieties of peppers are easier to grow for longer periods than others. Some peppers lose their leaves in the winter, and some don't (although how you treat them may or may not influence whether they lose their leaves). They say at least some habaneros don't lose their leaves in winter.
It seems to me that peppers that live longer probably produce better on the second and third years (but thereafter, I'm not so sure). I've heard this from two or three locations, but I don't have the sources, currently (I found them via web searches, probably in forums). However, I'm not referring to every pepper that can overwinter at least once as 'peppers that live longer'. Not all peppers that overwinter will thrive in this fashion, I hear.
Some people think most sweet peppers don't live that long, especially the hybrids, although if you take proper care of them, they might last somewhat longer and do better. However, I've read that at least big Bertha peppers, which are sweet, will do well on at least one subsequent year (in at least one person's experience).
I could be wrong, but it really seems like the peppers that overwinter indoors the best are those that tolerate cold the best. Even houses can get pretty cold for some peppers. So, that's good to take into consideration.
Peppers are notorious for dying when it gets cold. Most people think all peppers die at the slightest freeze. However, I have heard an account of someone in Beijing, China who had some wild peppers outside that survived snow.
Some of the more cold-tolerant peppers (and though they are all hot varieties, they just might survive mildly freezing temperatures) include these (although even the hardiest of these, with the possible exception of capsicum flexuosum, would probably die at 30 to 23°F (-1 to -5°C) and colder:
Of the cold-hardy hot peppers listed above, goats weed is the most closely related to sweet peppers. So, it might be possible to get a hybrid going there, with a certain amount of cold tolerance.
[Edit: Having grown red Peruvian Rocoto pepper plants, Black Cobra, and Chocolate Habanero, and observing them in a frost down at at least 28° F. I should note that each variety suffered from the frost. Some random plants didn't suffer much, though, but I'm not convinced it was due to cold-tolerance, since the same was true for other kinds of plants. In short, I'm not convinced they're any more cold-tolerant than other peppers. However, they do seem somewhat more cold-tolerant than my tomatoes, which suffered much more. Black Cobra is similar to Goat's Weed; so, that's why I mentioned it. I'm trying to start true Goat's Weed, now, though. So, maybe I'll see about it next year.]
I'm not sure how cold-hardy it is, but this very hot pepper also can be overwintered:
I'm guessing peppers that people call determinate may also be perennials, at least some of the time, and that they probably just have greater intervals without flowers (or else are better at flowering at other seasons than those people are normally used to). This is, however, just a guess. They could really die at the end of the season regardless of frost or care.
There do seem to be very few peppers that people say are determinate, with the exception of commercial field peppers (whatever varieties those are), but if you grow the seeds from a store pepper, the seedlings will probably resemble one of the parents of the hybrid instead of the hybrid itself.
Really, the only varieties of sweet pepper that I’ve specifically heard can overwinter indoors well (at least once) are big Bertha and potentially pepper 'Carmen' F1 (overwintered at least once). I've also heard of some orange bell peppers that overwinter well, but I don't know what varieties they are. More than one person has mentioned orange bell peppers in this regard.
Some people say pruning peppers is good (and some think it may be necessary to keep them alive), while others think this should not be done.
I've heard of a fertilizer (bloom builder plant food) that is supposed to help at least cayenne peppers to fruit well even in the middle of winter. That might work for sweet peppers, too.
I don't remember where I read it, but I hear nitrogen will help pepper plants to get big, while phosphorus will help them to develop fruit. Too much nitrogen may result in a huge plant without much fruit. However, if you're overwintering, that may be exactly what you want (a plant focusing on growing larger may thrive better than a plant focusing on growing fruit), for I hear you can switch the fertilizer later on to get more fruit after the plant has already gotten large. I think I read that you want equal parts nitrogen and phosphorus (or a little bit less phosphorus) if you want fruit, and less phosphorus (though I'm unsure how much less) if you want growth.
I've never heard of bell peppers (or any plants) dying if they don't get pollinated. The flowers may not turn into fruit, but that should be it. If you're worried about it, you can always remove the flowers. Some plants that are annuals, such as amaranth, will die after they flower and seed (and pruning off the flowers may extend the life of the plant somewhat, although it may not be aesthetically appealing). I don't believe this applies to peppers, but if there are determinate annual peppers, it just might be possible.
If I were experimenting with new peppers to see if they could live a long time and produce well, I would personally want to try peppers with one or more of these qualities first:
- Above average cold tolerance
- Takes a long time before it produces its initial batch of fruit
- Known to grow to be about 5' tall or more
- Orange color
- Hierloom or wild varieties
- Originate from South America or somewhere else warm.
I would also be more concerned with leaves than fruit until the plant got big. I would also want to keep the soil and air warm (they say if it's too cold for you, it's too cold for your peppers), and give it a nice grow light, such as this one (I haven't personally tried it, but I like the reviews).
If you happen to get a large and old pepper plant and are worried that it might die some day, I have some advice for you. Take cuttings and when they have roots, pot them. If you prune your pepper, you might as well try it with the clippings. Yes, you can take cuttings of peppers and root them (although they say it's somewhat more difficult than with tomatoes). Here's one way to do it (this way is said to work for many fruit trees and roses, too, although for roses I've also seen a recommendation to rip some of the skin off near the bottom, which might also work for other plants), and here's another, although if you have clippings to spare, I'd say test it with the plain water method too, just to see if it works (so you can save on hormone rooting powder). When you take cuttings, even if you have a hybrid pepper that doesn't produce seeds, you can still get more plants that produce exactly the same kind of pepper. People often recommend only taking small cuttings (for just about any kind of plant), but I would recommend experimenting beyond that advice if you have lots of clippings to try. When I did large cuttings with tomatoes, it worked well. Having roots that will be buried deep in the soil seems to be an advantage that makes it so you have to water less.
As the link above mentions, taking cuttings to grow and overwinter can sometimes be a better way to overwinter a pepper than digging it up.
Some of bell peppers would suit your purpose (greenhouse production of bell peppers uses indeterminate cultivars, but field pepper cultivars are determinate), according to this website and this one. If you stop pollinating, the plant will stop producing fruit, but it won't stop the vegetative growth. Indoors it is good to do this on an occasional regular basis, to give the plant a break period.
Indeterminate is what you want. Most people probably don't think of peppers as determinate and indeterminate (it's probably a tomato-thing to most people). Here's a forum post with people discussing it. Peppers and tomatoes are in the same family. At least some indeterminate peppers are supposed to be good for hydroponic gardening.
Indeterminate plants grow and produce fruit indefinitely, I hear. Determinate plants do for a season.
Here are some examples of indeterminate sweet peppers:
Growing your peppers indoors, I would recommend a grow light, if you want your peppers to flourish. I'm not really sure how well they do without them, but I'm about to find out, as I just planted some less than a week ago. Also, make sure they're not in a cold room. They are said to like warm soil. I notice my seed roots grow faster when the room is warm.
I've seen it written that you want at least five gallons of dirt, but that's more than square foot gardening would require (although some things, such as watermelon, prefer deep soil). If you have room and light (as well as pots), I'd use a bigger pot for bigger and stronger plants.