There is no EU passport (as the European Union is NOT a nation) but there is a common visa for the countries part of the Schengen agreement that you will have to apply to.

 

Passports quote the Country, and then if they are members of the EU.
For example:

But in those examples the passport also depends on the Schengen agreement, for cross boarder movements. This agreement has been signed by Switzerland for example (not a EU member) but not by the UK.