  We consider the family of CIFSs of generalized complex continued fractions
with a complex parameter space. This is a new interesting example to which we
can apply a general theory of infinite CIFSs and analytic families of infinite
CIFSs. We show that the Hausdorff dimension function of the family of the CIFSs
of generalized complex continued fractions is continuous in the parameter space
and is real-analytic and subharmonic in the interior of the parameter space. As
a corollary of these results, we also show that the Hausdorff dimension
function has a maximum point and the maximum point belongs to the boundary of
the parameter space.
