  We explore models of new physics that can give rise to large (100% or more)
enhancements to the rate of Higgs decay to $Z\gamma$ while still being
consistent with other measurements. We show that this is impossible in simple
models with one additional multiplet and also in well motivated models such as
the MSSM and folded SUSY. We do find models with several multiplets that carry
electroweak charge where such an enhancement is possible, but they require
destructive interference effects. We also show that kinematic measurements in
Higgs decay to four leptons can be sensitive to such models. Finally we explore
the sensitivity of four lepton measurements to supersymmetric models and find
that while the measurement is difficult with the high luminosity LHC, it may be
possible with a future high energy hadron collider.
