Are you doing anything to treat his fins? I ask because one of my males got the crap kicked out of him today and I am looking for every bit of advice I can to help heal him.
This a bit off topic, but Melafix will work wonders for your problem, it can be found in pretty much every petstore that sells fish supplies, but it is often much, much cheaper online, I know of a few website where you can get a 16oz bottle for around $7, they usually range about $14-$16 in the store for the same size bottle. I have had good luck nursing fish back to health with this medication.
Now for the OP as to min depth for a betta fish, I suppose you can keep it in a dish with water barely enough to cover it and it will survive. Bettas often get abused as pets because they can breath air, so many people in the hobby think they do not require as great of care as other fish and treat them as such, which isn't true. You should provide you Betta with a filter, heater if its cold in the room with the tank, plenty of space, they need space to swim, and you should also try and provide your betta with a couple of hiding spots that it can pick and choose from, plants with large sturdy leaves work well, often times a betta will use a large leaf as a "hammock". I personally would never keep
any fish in a aquarium smaller than
ten gallons, to me its just cruel to expect a fish to live its entire life in 1/2 gallon of water and be "happy", which is very different than simply being alive. Other people will disagree with me stating that in the wild bettas can often be found in small pools of very dirty water, but it has always been my practice in the aquarium hobby to make the best environment that mimics nature, not to emulate it to the extent that is reduces the quality of life for the fish.
