  The article studies a generalization of the elliptic billiard to the complex
domain. We show that the billiard orbits also have caustics, and that the
number of such caustics is bigger than for the real case. For example, for a
given ellipse E, there exist exactly two confocal ellipses such that the
triangular orbits of E are circumscribed about one of them, and each tangent
line to one of those ellipses is a side of a triangular orbit. We also give a
bound on the number of caustics for orbits with a fixed number of sides.
