  We consider a family of CIFSs of the generalized complex continued fractions
with a complex parameter space. We show that for each CIFS of the family, the
Hausdorff measure of the limit set of the CIFS with respect to the Hausdorff
dimen/sion is zero and the packing measure of the limit set of the CIFS with
respect to the Hausdorff dimension is positive (main result). This is a new
phenomenon of infinite CIFSs which cannot hold in finite CIFSs. We prove the
main result by showing some estimates for the unique conformal measure of each
CIFS of the family and by using some geometric observations.
