  We examine a pair of dynamical systems on the plane induced by a pair of
spanning trees in the Cayley graph of the Super-Apollonian group of Graham,
Lagarias, Mallows, Wilks and Yan. The dynamical systems compute Gaussian
rational approximations to complex numbers and are "reflective" versions of the
complex continued fractions of A. L. Schmidt. They also describe a reduction
algorithm for Lorentz quadruples, in analogy to work of Romik on Pythagorean
triples. For these dynamical systems, we produce an invertible extension and an
invariant measure, which we conjecture is ergodic. We consider some statistics
of the related continued fraction expansions, and we also examine the
restriction of these systems to the real line, which gives a reflective version
of the usual continued fraction algorithm. Finally, we briefly consider an
alternate setup corresponding to a tree of Lorentz quadruples ordered by
arithmetic complexity.
