I cut a piece of basil and put it inside a vase originally for pure decorative purposes but now I see that it has started to grow roots and it still looks relatively healthy to me. I change the water of the vase every other day for about a week now and the plant's roots keep on growing. Should I keep it in its vase and just change water every now and then or would it be wiser to try planting it?
Definitely the plant needs nutrients (soil or fertilizers). Note there are some liquid fertilizers that can be applied on leaves.
On the other hand, I don't know if such plant will survive on planting it in soil: the roots are very long, so you need a large pot or hole, but the plant is not so large, so the risk that the roots are in the wrong place is high (and also the risk to damage them).
So, it is you call. Possibly it will not be a good plant to eat (variety), and IMHO it cannot survive much more time without fertilizers.
I've grown basil hydroponically using aquarium water for the nutrients and it looked a lot better than yours which needs a lot more sunlight. In fact most of the basil I see in the supermarket seems to be grown hydroponically.
If you don't have any hydroponic mixes, then just chuck some garden soil in the bottom. And change the water as you are but put it on a sill with lots of sunlight.
Ps: healthy hydroponic plants have huge roots, often more than the plant above the roots.
Beautiful idea using this bulb. Where did you find this incandescent bulb? An antique store?? Grins. You need to know that water roots are not at all the same as soil roots. You could easily vegetatively reproduce this plant just by cutting off one or two leaves, use rooting powder (I have finally been convinced until more research)...this rooting chemistry (gel or powder) seems to really make a difference with success. You could plant this guy in soil and cross your fingers. People experiment with avocados rooting them first in water then planting that seed in soil and most avocados seem to do well if other aspects such as no over watering, not using garden soil just sterilized potting soil...are considered.
My experience shows plants when young or trying to establish themselves within some niche should not have added fertilizer until...the 3rd set of leaves. Have you added any fertilizer to this light bulb's water? Fertilizer is critical after the plant makes it through the hurdles of becoming established. No compost, no sand, no additives in the potting soil at all including water holding gimmicks such as gels and sponges and most certainly no added fertilizer. You need to be in full control of that environment's chemistry. Too many just throw a little of this, a little of that still thinking if someone says it helps it must be true. KISS while a novice, to enjoy more success, is a great motto! (Keep It Simple Stupid) and don't you take that as a personal anything. I find when I get to busy to think this is my fall back! Basil is a great plant to grow indoors. Try Osmocote extended release 14-14-14 after transplanting. Only once per season. Then forget about adding anything other than water for awhile. Mycorrhizae is also helpful for high stress situations for plants. Are you using tap water? Check out our section on potting, pots, soils and indoor plants or plants in pots.
Looking more closely at your basil I know it needs more light. There are the coolest grow lights on the market right now and a huge variety of styles. We are researching the LED's...we've got HPS and metal halide. To have a successful indoor 'garden'...herbs and lettuce work well you should look into a real grow light. For a small area 3'X3' T-5's fluorescents are very inexpensive. There is so much new stuff in lighting it will be awhile before I am able to synthesize and experiment to offer better advice.