  In the Next-to-Minimal Supersymmetric Standard Model (NMSSM) a variety of
parameter configurations yields a Higgs boson consistent with the one observed
at the LHC. Additionally, the Higgs sector of the model can contain explicit
CP-violating phases even at the tree level, in contrast with the Minimal
Supersymmetric Standard Model (MSSM). In this article we present the one-loop
Higgs boson mass matrix of the complex NMSSM in the
renormalisation-group-improved effective potential approach. We also present
the trilinear Higgs boson self-couplings as well as various partial decay
widths of a generic CP-mixed Higgs boson in the model. We then analyse a very
interesting phenomenological scenario wherein the decay of a relatively light
pseudoscalar-like Higgs boson into ~ 125 GeV SM-like Higgs boson(s) is induced
by non-zero CP-violating phases. We discuss in detail a few benchmark cases in
which such a decay can contribute significantly to the production of SM-like
Higgs bosons at the LHC on top of the gluon fusion process. It can thus be
partially responsible for the gamma.gamma excess near 125 GeV due to the
subsequent decay of the SM-like Higgs boson. Such a scenario is extremely
difficult to realize in the complex MSSM and, if probed at the LHC, it could
provide an indication of the non-minimal nature of supersymmetry.
