What if the mother, because our relation ship can't work out decides to move out and say the baby is living with her. Can the father fight for the baby and he has a better paying job and has the Want to take care of it? Also does he stand a good chance?

A father has the right to file for custody or joint custody if the mother has a good job and is rich,

Fathers in the US are more and more getting at least joint custody if they file and fight for it.

Courts still tend to giving preference in primary custody to a mother. But times are changing. At the very least, you can get joint custody and frequent visitation.