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In the Next-to-Minimal Supersymmetric Standard Model (NMSSM) a variety ofparameter configurations yields a Higgs boson consistent with the one observedat the LHC. Additionally, the Higgs sector of the model can contain explicitCP-violating phases even at the tree level, in contrast with the MinimalSupersymmetric Standard Model (MSSM). In this article we present the one-loopHiggs boson mass matrix of the complex NMSSM in therenormalisation-group-improved effective potential approach. We also presentthe trilinear Higgs boson self-couplings as well as various partial decaywidths of a generic CP-mixed Higgs boson in the model. We then analyse a veryinteresting phenomenological scenario wherein the decay of a relatively lightpseudoscalar-like Higgs boson into ~ 125 GeV SM-like Higgs boson(s) is inducedby non-zero CP-violating phases. We discuss in detail a few benchmark cases inwhich such a decay can contribute significantly to the production of SM-likeHiggs bosons at the LHC on top of the gluon fusion process. It can thus bepartially responsible for the gamma.gamma excess near 125 GeV due to thesubsequent decay of the SM-like Higgs boson. Such a scenario is extremelydifficult to realize in the complex MSSM and, if probed at the LHC, it couldprovide an indication of the non-minimal nature of supersymmetry.

Novel Higgs-to-125 GeV Higgs boson decays in the complex NMSSM