  Measuring the Higgs couplings accurately at colliders is one of the best
routes for finding physics Beyond the Standard Model (BSM). If the measured
couplings deviate from the SM predictions, then this would give rise to
energy-growing processes that violate tree-level unitarity at some energy
scale, indicating new physics. In this paper, we extend previous work on
unitarity bounds from the Higgs potential and the Higgs couplings to vector
bosons and the top quark; to the Higgs couplings to $\gamma\gamma$ and $\gamma
Z$. We find that while the HL-LHC might be able to find new physics in the
$\gamma Z$ sector, the scale of new physics in both sectors is mostly beyond
its reach. However, accurate measurements of the leading couplings of the two
sectors in the HL-LHC can place stringent limits on both the scale of new
physics and on other Higgs couplings that are difficult to measure. In
addition, the scale of new physics is mostly within the reach of the $100$ TeV
collider.
