How to bring one's PhD research to another university?
You are free to take your idea wherever you would like. You cannot take your professor's project proposal to another university without her permission.

The professor at your school seeing this as a "betrayal" is a separate issue. A good advisor will of course want a good student to stick around. A good mentor will want what's best for the student—regardless of what's in the mentor's best interests as a potential advisor. 

So, my advice is just to talk to the professor and let him know that you'd like to apply at other universities as well. (It's your right to do so, of course.) Ask if the professor would be willing to write letters supporting you. If the professor isn't supportive of this, you'll have your answer. (And if the professor is not supportive, then do not get letters of recommendation from that professor!)