  We present a property satisfied by a large variety of complex continued
fraction algorithms (the "finite building property") and use it to explore the
structure of bijectivity domains for natural extensions of Gauss maps.
Specifically, we show that these domains can each be given as a finite union of
Cartesian products. In one complex coordinate, the sets come from explicit
manipulation of the continued fraction algorithm, while in the other coordinate
the sets are determined by experimental means.
