  Many beyond the Standard Model (BSM) scenarios involve Higgs couplings to
additional electroweak fields. It is well established that these new fields may
modify Higgs gamma-gamma and gamma-Z decays at one-loop. However, one
unexplored aspect of such scenarios is that by electroweak symmetry one should
also expect modifications to the Higgs Z-Z coupling at one-loop and, more
generally, modifications to Higgs production and decay channels beyond
tree-level. In this paper we investigate the full BSM modified electroweak
corrections to associated Higgs production at both the LHC and a future lepton
collider in two simple SM extensions. From both inclusive and differential NLO
associated production cross sections we find BSM-NLO corrections can be as
large as O(>10%) when compared to the SM expectation, consistent with other
precision electroweak measurements, even in scenarios where modifications to
the Higgs diphoton rate are not significant. At the LHC such corrections are
comparable to the involved QCD uncertainties. At a lepton collider the Higgs
associated production cross section can be measured to high accuracy (O(1%)
independent of uncertainties in total width and other couplings), and such a
deviation could be easily observed even if the new states remain beyond
kinematic reach. This should be compared to the expected accuracy for a
model-independent determination of the Higgs diphoton coupling at a lepton
collider, which is O(15%). This work demonstrates that precision measurements
of the Higgs associated production cross section constitute a powerful probe of
modified Higgs sectors and will be valuable for indirectly exploring BSM
scenarios.
