Modified Higgs branching ratios versus CP and lepton flavor violation

New physics thresholds which can modify the diphoton and dilepton Higgs branching ratios significantly may also provide new sources of CP and lepton flavor violation. We find that limits on electric dipole moments impose strong constraints on any CP-odd contributions to Higgs diphoton decays unless there are degeneracies in the Higgs sector that enhance CP-violating mixing. We exemplify this point in the language of effective operators and in simple UV-complete models with vector-like fermions. In contrast, we find that electric dipole moments and lepton-flavor-violating observables provide less stringent constraints on new thresholds contributing to Higgs dilepton decays.