Can I bring my girlfriend when I move into my new house?

This is ok. As long as you enter the new house, you can definitely bring your girlfriend directly.

If you have a particularly good relationship with your girlfriend, or if you and your girlfriend are moving towards the goal of getting married, I think you can definitely bring your girlfriend directly. Only in this way can living in this new house be meaningful or happy.

So at this time, you must take a look at your own ideas. If you really want your girlfriend to move in with you, you can definitely get her permission.