  This paper provides some preliminary results on the dynamics of certain
complex continued fractions. After establishing some general number theoretic
results, we explore the dynamics of the natural extension map associated to a
specific complex continued fraction algorithm (the "diamond" algorithm). We
prove that this map has a bijectivity domain that is a subset of a trapping
region for the map and, moreover, that both these sets have a "finite product
structure" arising from a finite partition specific to the particular
algorithm.
