My car keeps dying with a new battery - could it be the alternator?

I assume that by “dying,” you mean you are getting no crank when you turn the key. Or will it shut down when you are driving too?

You should have your alternator and that battery checked. (New ones can have defects). Someone should also check your main power cables and their connections. But mostly someone should check it for parasitic drains - something sucking power that it shouldn’t be when the car is off.