Bitcoin was created as an alternative to traditional money.
I can imagine the future where BTC becomes not just the most popular, but also fully accepted alternative to fiat. And yet, it will be just an alternative.
In line with your thoughts I am thinking that bitcoin will be the official currency and will be accepted worldwide. But it only happens online.
Bitcoin will be the official representative of dollars in the world of Virtual currency. If you have a 100 dollars in the bank the bank will deposit or transfer bitcoins equivalent to your 100 dollars. If you transfer your bitcoins to somebody the bank will also reduce your dollars and transfer it to that someone you sent.