Of course bank will ever exist. Bank are not evil. The system is.
Bank will exist for ever, if tomorrow something REALLY serious open door who can keep your bitcoin safe with insurance + make you able to spend it pretty easily with multiples account (saving, current) protected by different password + make you able to invest in stock or whatever or can do it for you, a trader will take your bitcoin and play with it and take a comission (because you don't have time to take care it).
I will sign up and pay for it for sure (in a reasonnable way) Maybe i will not put everything i have, because i'm paranoid. But bank will forever exist because security is a pro thing, and investment is needed in our economy. Maybe is more risky for me to take care about my own security than give it to a bank. Really i'm not really good at security, some of you are, but we are not all engineer with 4 lapops and 2 desk computer at home.
Bank is needed overall for investment, i believe about p2p stocks market, but i think it's will be like a "2nd market" for smaller company, we need also centralised. We need p2p and centralised, we need to have the choose. Serious company with real invest they need some centralisation.
I hope people with large amount of BTC will realise they better to invest Bitcoin in Bitcoin company instead of keep it in cold storage, this will not help Bitcoin to be what it need to be
But people with millions of dollars worth bitcoin they need some insurance.
I hope a Bitcoin trust fund will emerge soon. Like they take the top10 of best startup bitcoin in different localisation to mitige the risk. You put an invest of 5 bitcoin for example and you receive 10 different colored coin from company. I feel like we need kind of centralised organisation, or it's will be a giant mess, because you need a huge investment to make a system like this. P2P is very good, but i will not rules the world. We need some "centralised" organisation.
Centralised not mean dark organisation, Bitcoin allow transparency and all company who want to be trusted will need to open their account to public soon. Its start to happening. Maybe bank of tomorrow will not be Bank of today, it's all depends how they handle their transition in this new system. It's like paypal i feel, maybe Paypal tomorrow will be stronger then never if they integrate Bitcoin in a right way. Maybe they will not and will disapear in 5-10years.
I'm sure pretty sure we will have a giant schism in banking system. Some will fight bitcoin to dead to survive, and some will try to transition into, they will invest and fund Bitcoin bank.
I think bank who will try to fight BTC are dangerous and maybe they can make some damage to Bitcoin, maybe with politics, maybe with misinformation, maybe with illegal thing, maybe all of 3. But they will not destroy the idea, so the fight is already lost, they can just try to delay it like 5-10years.
They can be sucesfull in transition in a new system, dont forget Banker are kind of genious (they recruit something like 30% of all the better student in the world, and they are liberalist by education so bitcoin is a dream who can come true for them?), have kind of illimited fund and it's an intellectual elite of this world, dont underestimate this people lol... If they see an opportunity, they will take it seriously. Bank will continue exist of course, they have so much money, they just can't lose 100%.
Centalised is good because centralised organisation is more easy to make profit, so they can invest massively, and can do faster and better what p2p have hard time to make. It's a reality. It's how the world works for thousand of years, and will continue i believe.