If $O$ is the ring of integers in a number field, $PSL_n(O)$ is almost always generated by its elementary matrices. The only exceptions are when $n=2$ and $O$ is non-Euclidean, imaginary quadratic. In these cases, group presentations are computed algorithmically. There is no known, general formulation. That these algorithms eventually terminate (and when they terminate) follows from a bound on the complexity of a Ford domain associated to $PSL_2(O)$. We give a new upper bound on this complexity that comes within a power of $\log|\Delta|$ of the value in question, where $\Delta$ is the discriminant of $O$. This improves on the previous best-known upper bound by a factor of at least $\Delta^2$.