Can there be dollar version of bitcoin?
That would be absolutely possible. In fact, there is a cryptocurrency out there called "Hashdollar" which is trying just that. I didn't look into it yet but if you tie the value of the Hashdollar to the US Dollar, you could use the p2p network to make "real" dollar transaction. The problem here is though, who will guarantee for the value of the Hashdollar? The value of Bitcoin is mostly made by market mechanisms but if you tie the value of a crypto currency to the dollar, you would need to have some authority to guarantee for that value and I do not see that happening. 

Of course the feds could just come up with their own version of a cryptocurrency (eDollar?) and this might actually happen very soon.

Hashdollar: https://bitcointalk.org/index.php?topic=368237.20