You wouldn't want to file identical claims. Identical claims will receive a "Statutory Double Patenting" rejection.
On other note why to pay fee twice on claim already filed and pending.
Unlike the "Non-Statutory Double Patenting" rejection referred to by Gary, the you can not overcome the Statutory Double Patenting rejection with a terminal disclaimer. You must amend the claims. See MPEP 804.02(I).
Understand that in order to file a continuing application, the originally filed application, or an continuation/divisional/continuation-in-part based on the originally filed application must be pending (i.e., not abandoned, not issued).
If you are filing another application as a result of an examiner's requirement for restriction, the application would be called a divisional application rather than a continuing application. Statutory double patenting is prohibited in divisional applications resulting from an examiner's requirement for restriction. See MPEP 804.01.