The other type of account, was called a "fixed term account", where you lent your money to the bank for a specified amount of time. These accounts did pay compound interest because the bank could lend out your money at interest.
The only reason to use the first type of account, was for keeping your money safe, or facilitating transactions. Under a bitcoin monetary system, there would be no need to put your bitcoins in a non-interest paying "demand account".
The second type of account will continue to exist under a bitcoin monetary system. People who would like to earn interest on their bitcoins, will give them to a bank to lend out. The bank will make their profits by the difference between the lending and deposit interest rates.
However, depositors earning interest in a bitcoin bank, will carry more risk than in a fiat system, because if the bank makes losses on their loan book, they will have to pass that loss on to depositors, if they dont have enough capital to absorb the losses themselves.
A central bank will not be able to rescue an insolvent bitcoin bank, because the central bank will not have any special ability to create bitcoins or access credit in bitcoins.
Depositors in a bitcoin bank should do their homework and make sure the bank is well capitalised. If not they should purchase deposit insurance.
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