The same action can be constructed directly from D, by having GL(2, D) -> PGL(2, C) act on Minkowski space and hence on hyperbolic 3-space. This is an example of an arithmetic Kleinian group. Up to finite index considerations, it is a Bianchi group.
Bianchi group actions on hyperbolic 3-space are well-studied, as described in Subsection 2.2 of https://arxiv.org/pdf/1204.6697 , see especially Figure 1 which shows a fundamental domain. I wonder whether the literature on fundamental domains of Bianchi group actions vindicates James Dolan's intuition that they are "slightly stubby voronoi regions for the quadratic integers in D."
A Bianchi group action on hyperbolic 3-space always has at least one cusp. (In fact, the set of cusps is in bijection with the class group of the imaginary quadratic field, see https://en.wikipedia.org/wiki/Bianchi_group ) Since the Bianchi group acts via isometries, the presence of a cusp forces orbits to contain many points which lie on one horosphere.
Generalizing to abelian surfaces which are isogenous to E x E only introduces finite index complications, because the NS group is still a rank 4 lattice. But generalizing any further will decrease the rank of the NS group, by the MO question linked above.
There is a cool-looking paper which discusses this construction and claims to use hyperbolic group actions to prove a theorem about the nef cone: https://arxiv.org/pdf/0901.3361 (see discussion here https://mathoverflow.net/a/27353 ). I haven't looked in detail though