  We study exotic Higgs decays $h \to Z X$, with $X$ an invisible beyond the
Standard Model (SM) particle, resulting in a semi-dark final state. Such exotic
Higgs decays may occur in theories of axion-like-particles (ALPs), dark photons
or pseudoscalar mediators between the SM and dark matter. The SM process $h\to
Z\nu\bar{\nu}$ represents an irreducible "neutrino floor" background to these
new physics searches, providing also a target experimental sensitivity for
them. We analyze $h \to Z + \text{invisible}$ searches at the LHC and a future
ILC, showing that these exotic Higgs decays can yield sensitivity to unexplored
regions of parameter space for ALPs and dark matter models.
