  We present a general method for calculating the leading contributions to h ->
gg and h -> gamma gamma in models where the Higgs weakly mixes with a nearly
supersymmetric extra sector. Such mixing terms can play an important role in
raising the Higgs mass relative to the value expected in the MSSM. Our method
applies even when the extra sector is strongly coupled, and moreover does not
require a microscopic Lagrangian description. Using constraints from holomorphy
we fix the leading parametric form of the contributions to these Higgs
processes, including the Higgs mixing angle dependence, up to an overall
coefficient. Moreover, when the Higgs is the sole source of mass for a
superconformal sector, we show that even this coefficient is often calculable.
For appropriate mixing angles, the contribution of the extra states to h -> gg
and h -> gamma gamma can vanish. We also discuss how current experimental
limits already lead to non-trivial constraints on such models. Finally, we
provide examples of extra sectors which satisfy the requirements necessary to
use the holomorphic approximation.
