  The interpretation of Higgs data is typically based on different assumptions
about whether there can be additional decay modes of the Higgs or if any
couplings can be bounded by theoretical arguments. Going beyond these
assumptions requires either a precision measurement of the Higgs width or an
absolute measurement of a coupling to eliminate a flat direction in precision
fits that occurs when $|g_{hVV}/g_{hVV}^{SM}|>1$, where $V=W^\pm, Z$. In this
paper we explore how well a high energy muon collider can test Higgs physics
without having to make assumptions on the total width of the Higgs. In
particular, we investigate off-shell methods for Higgs production used at the
LHC and searches for invisible decays of the Higgs to see how powerful they are
at a muon collider. We then investigate the theoretical requirements on a model
which can exist in such a flat direction. Combining expected Higgs precision
with other constraints, the most dangerous flat direction is described by
generalized Georgi-Machacek models. We find that by combining direct searches
with Higgs precision, a high energy muon collider can robustly test single
Higgs precision down to the $\mathcal{O}(.1\%)$ level without having to assume
SM Higgs decays. Furthermore, it allows one to bound new contributions to the
width at the sub-percent level as well. Finally, we comment on how even in this
difficult flat direction for Higgs precision, a muon collider can robustly test
or discover new physics in multiple ways. Expanding beyond simple coupling
modifiers/EFTs, there is a large region of parameter space that muon colliders
can explore for EWSB that is not probed with only standard Higgs precision
observables.
