Going elsewhere for an M.S. in Computer Science?
Yes, preferably go somewhere else for the M.S. if what you want is the M.S. degree.
The benefits of 'new challenges' (already mentioned) are worth it.

However, if your goal is to be admitted for PhD, then the research experience that you could get ahead of applying for PhD will be more valuable than the M.S. degree. I've seen PhD applicants with stronger research experience and having BS degree only, than other applicants that have a MS. 
If you can start having conversations with faculty at your institution with whom you'd like to do research, then you'll be able to make a better decision. Just be clear with them saying that you want learn and do research to have a better chance for PhD admission at other schools two years down the line.