We show that supersymmetric "new physics" beyond the MSSM can naturally
accommodate a Higgs mass near 126 GeV and enhance the signal rate in the Higgs
to diphoton channel, while the signal rates in all the other Higgs decay
channels coincide with Standard Model expectations, except possibly the Higgs
to Z-photon channel. The "new physics" that corrects the relevant Higgs
couplings can be captured by two supersymmetric effective operators. We provide
a simple example of an underlying model in which these operators are
simultaneously generated. The scale of "new physics" that generates these
operators can be around 5 TeV or larger, and outside the reach of the LHC.Comment: 24 pages, 4 figure