We are trying to grow purple potatoes for the first time, and the plants look really nice. I am wondering if harvesting them is any different than for white potatoes?
Purple potatoes are fun. I don't believe there is any real difference harvesting them from white-fleshed ones.
I don't know how you are growing them but when I've done potatoes in the past I have tried mounding up soil and also spent hay and straw (hay/straw was easier for harvesting because there's no digging at harvest time and it's easier to just break open a bale and cover them.).
I just keep mounding up the material over them, leaving just the topmost section exposed to the sunlight. Eventually they stopped growing so much vertically and flowered (nice looking flowers, at least I think so) and then the plant top growth died off. At some point after that it is fine to harvest them. The flowering is an indicator that you could harvest. Well, technically you could try to harvest whenever but your mileage may vary. I have harvested early when I needed potatoes and got some though they were smaller. Typically I waited for the tops to die off a bit.