  For the search for additional Higgs bosons in the Minimal Supersymmetric
Standard Model (MSSM) as well as for future precision analyses in the Higgs
sector a precise knowledge of their production properties is mandatory. We
evaluate the cross sections for the neutral Higgs boson production at e^+e^-
colliders in the MSSM with complex parameters (cMSSM). The evaluation is based
on a full one-loop calculation of the production channels e^+e^- -> h_i Z, h_i
gamma, h_i h_j (i,j = 1,2,3), including soft and hard QED radiation. The
dependence of the Higgs boson production cross sections on the relevant cMSSM
parameters is analyzed numerically. We find sizable contributions to many cross
sections. They are, depending on the production channel, roughly of 10-20% of
the tree-level results, but can go up to 50% or higher. The full one-loop
contributions are important for a future linear e^+e^- collider such as the ILC
or CLIC. There are plans to implement the evaluation of the Higgs boson
production cross sections into the code FeynHiggs.
