  The BMSSM framework is an effective theory approach that encapsulates a
variety of extensions beyond the MSSM with which it shares the same field
content. The lightest Higgs mass can be much heavier than in the MSSM without
creating a tension with naturalness or requiring superheavy stops. The
phenomenology of the Higgs sector is at the same time much richer. We
critically review the properties of a Higgs with mass around 125GeV in this
model. In particular, we investigate how the rates in the important inclusive 2
photons channel, the 2 photons + 2 jets and the ZZ to 4 leptons (and/or WW) can
be enhanced or reduced compared to the standard model and what kind of
correlations between these rates are possible. We consider both a vanilla model
where stops have moderate masses and do not mix and a model with large mixing
and a light stop. We show that in both cases there are scenarios that lead to
enhancements in these rates at a mass of 125GeV corresponding either to the
lightest Higgs or the heaviest CP-even Higgs of the model. In all of these
scenarios we study the prospects of finding other signatures either of the
125GeV Higgs or those of the heavier Higgses. In most cases the $\oo{\tau}\tau$
channels are the most promising. Exclusion limits from the recent LHC Higgs
searches are folded in our analyses while the tantalising hints for a Higgs
signal at 125GeV are used as an example of how to constrain the BMSSM and/or
direct future searches.
