  The Next-to-Minimal Supersymmetric Extension of the Standard Model (NMSSM)
with a Higgs sector containing five neutral and two charged Higgs bosons allows
for a rich phenomenology. In addition, the plethora of parameters provides many
sources of CP violation. In contrast to the Minimal Supersymmetric Extension,
CP violation in the Higgs sector is already possible at tree-level. For a
reliable understanding and interpretation of the experimental results of the
Higgs boson search, and for a proper distinction of Higgs sectors provided by
the Standard Model or possible extensions, the Higgs boson masses have to be
known as precisely as possible including higher-order corrections. In this
paper we calculate the one-loop corrections to the neutral Higgs boson masses
in the complex NMSSM in a Feynman diagrammatic approach adopting a mixed
renormalization scheme based on on-shell and $\bar{DR}$ conditions. We study
various scenarios where we allow for tree-level CP-violating phases in the
Higgs sector and where we also study radiatively induced CP violation due to a
non-vanishing phase of the trilinear coupling $A_t$ in the stop sector. The
effects on the Higgs boson phenomenology are found to be significant. We
furthermore estimate the theoretical error due to unknown higher-order
corrections by both varying the renormalization scheme of the top and bottom
quark masses and by adopting different renormalization scales. The residual
theoretical error can be estimated to about 10%.
