  The coincidence site lattices (CSLs) of prominent 4-dimensional lattices are
considered. CSLs in 3 dimensions have been used for decades to describe grain
boundaries in crystals. Quasicrystals suggest to also look at CSLs in
dimensions $d>3$. Here, we discuss the CSLs of the root lattice $A_4$ and the
hypercubic lattices, which are of particular interest both from the
mathematical and the crystallographic viewpoint. Quaternion algebras are used
to derive their coincidence rotations and the CSLs. We make use of the fact
that the CSLs can be linked to certain ideals and compute their indices, their
multiplicities and encapsulate all this in generating functions in terms of
Dirichlet series. In addition, we sketch how these results can be generalised
for 4--dimensional $\Z$--modules by discussing the icosian ring.
