More of a comment than an answer. Chapter 5 of William Goldman's book Complex Hyperbolic geometry' has a great detail of information on the bisectors in complex hyperbolic space (so real dimension 2n, n>1) and has many pictorial representations of them. They are beautiful objects. Goldman asserts that there are no real codimension 1 totally geodesic submanifolds in complex hyperbolic space so the bisectors cannot be totally geodesic. The bisectors of complex hyperbolic space are minimal surfaces, all congruent to each other.