I think it's a fantastic idea. The bank itself would need a fairly substantial capital backing when it first opens, in BTC and probably a "real" currency like USD. I'm very interested in this and if anyone has the programming skills to create such a thing, I'd be happy to pay you, in American greenbacks.
As stated earlier, I'm seriously thinking and exploring the idea... As a business man, however the first thing is to have a concrete business plan.
As such that is what I'm doing with now, trying to balance the potential books, how much investment, how to generate profits, share-holders or not?, advantage to users, etc.
Ironically given the current state of main economy banks, banking is all about trust... And that is a hard thing to get, for a bank to succeed, at the beginning it has to inspire confidence that it can keep you money safe and knows how to grew it (for interest and share-holders).
One possibility I've considered is to follow the model of the co-operative banks (AKA building societies) in the UK. On these there are no share-holders or owners except the customers. Your deposit with the bank, entitles you to a dividend each years based on the profits made.