On that fancy mixer in question, it looked like a spiral hook in one of their pictures. With a hook of this type the dough will not climb up the hook, instead, it will be constantly be pushed down toward the bottom of the bowl. As for mixing the dough, a pizza dough only needs to be mixed to a point where it has a smooth, satiny appearance. Mixing beyond this is not necessary or desirable unless you're looking for a crumb structure that more closely resembles bread than pizza.
Tom Lehmann/The Dough Doctor