For real-valued continued fractions, the natural extension of the Gauss map has a global attractor with a simple structure coming from a "cycle property". Because this cycle structure is strictly one-dimensional, the "finite building property" was developed as an alternative to analyze /complex/ continued fraction algorithms. For algorithms with this property, the domain in $\mathbb C\times\mathbb C$ of the natural extension of the continued fraction map can be described 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.