  We analyse the impact of explicit CP-violation in the Higgs sector of the
Next-to-Minimal Supersymmetric Standard Model (NMSSM) on its consistency with
the Higgs boson data from the Large Hadron Collider (LHC). Through detailed
scans of the parameter space of the complex NMSSM for certain fixed values of
one of its CP-violating (CPV) phases, we obtain a large number of points
corresponding to five phenomenologically relevant scenarios containing $\sim
125$ GeV Higgs boson(s). We focus, in particular, on the scenarios where the
visible peaks in the experimental samples can actually be explained by two
nearly mass-degenerate neutral Higgs boson states. We find that some points
corresponding to these scenarios give an overall slightly improved fit to the
data, more so for non-zero values of the CPV phase, compared to the scenarios
containing a single Higgs boson near 125 GeV.
