Question: Will I need a Scottish passport?
Answer: Scottish citizens will be able to apply for a Scottish passport from day one of independence, with a Scottish passport costing the same as a UK passport. On the other hand, people will be able to continue using their existing UK passport until it expires. They can then choose to get a Scottish passport when their existing passport is due for renewal. Yes Scotland 2014
Remember also that the Scottish Government will allow dual citizenship. It will be for the rest of the UK to decide whether it allows dual UK/Scottish citizenship - though we know that dual citizenship is allowed for those with any other nationality, and the UK Government has said that "it is likely that it would be possible for an individual to hold both British and Scottish citizenship".
So it will also likely be possible to buy a replacement British passport - either in addition to a Scottish passport or as an alternative.
Scotland’s Future explains that we will remain part of the Common Travel Area between the nations of the British Isles. Being part of the Common Travel Area means people won’t need a passport to travel to the rest of the UK or Ireland.