Would it be possible to start a global nonprofit bank, and would anyone be interested in that?

This idea really just popped into my head and is very half-baked. Think Bitcoins but we're not trying to replace the dollar, we're trying to replace the way personal banking gets done. What would you want out of a bank? What types of things would make it beneficial to all? What would it take to make sure that it was completely secure and safe? Is this just a sophomoric idea or is there potentially something to this?