The ring of Hurwitz integers, being both a left and a right PID, could be thought of as being arithmetically fairly simple. However, the fact that it is not commutative entails some complications, but also some surprises, as well as some interesting open problems. In this talk we will describe Conway and Smith metacommutation problem, and some almost forgotten results on the related ring of Lipschitz integers. We will then present some results, obtained in joint work with Luis Roçadas, on some relationships between the arithmetic and the geometry of Lipschitz integers, namely certain divisibility relations between a given Lipschitz integer and some other integers built from it using the vector product. Some speculations on a possible integer factorization method involving quaternion arithmetic will also be presented.