Home Hydroponics: Yes, You Can!
Get yourself a pot with a water reservoir (or make your own), some substrate material (I use perlite), and figure out what you will use for nutrient solution.
I use a commercial mix from the local hydroponic store (ignore all the mixes about "Big Buds", sigh-- they are not for veggies, and they are high-nitrogen anyway so you would get more leaves than fruit). Dry mixes are best, followed by concentrated liquid mixes that you dilute. Don't bother with a premixed solution, you are paying a lot for water!
You might be able to use a combination of conventional minerals, like dusting greensand into the medium, and some bone meal, and then using an off the shelf fertilizer like VF-111 or a concentrated fish emulsion (Alaska, Atlas). I haven't really tried that yet, since the little container of the dry hydroponic mix I have has lasted 3 years already for me, with only a few 4-foot long planters a year and a teaspoon of mix into each weekly. As you can see, it grew some nice eggplants for me! Thai Lavender (long) and Fairy Tale (short, variegated)
I recommend reading up a bit on the net on hydroponics. It's really pretty simple if you are doing it at home, rather than trying to automate it in a commercial greenhouse to produce bumper crops at timed intervals. Sure, if you get the mix too weak, your peppers might take an extra week or two to ripen. No big deal at home, a real big deal if you have a quarter-acre of them in hoop-row greenhouses and a contract to deliver them to some restaurant chain. ;-)