  If the soft Supersymmetry (SUSY) breaking masses and couplings are complex,
then the associated CP-violating phases can in principle modify the known
phenomenological pattern of the Minimal Supersymmetric Standard Model (MSSM).
We investigate here their effects on Higgs boson production in the gluon-gluon
fusion mode at the Tevatron and the Large Hadron Collider (LHC), by taking into
account all experimental bounds available at present. The by far most stringent
ones are those derived from the measurements of the Electric Dipole Moments
(EDMs) of fermions. However, it has recently been suggested that, over a
sizable portion of the MSSM parameter space, cancellations among the SUSY
contributions to the EDMs can take place, so that the CP-violating phases can
evade those limits. We find a strong dependence of the production rates of any
neutral Higgs state upon the complex masses and couplings over such parts of
the MSSM parameter space. We show these effects relatively to the ordinary MSSM
rates as well as illustrate them at absolute cross section level at both
colliders
