  It is often stated that elimination of tree-level flavor-changing neutral
currents in multi-Higgs models requires that all fermions of a given charge to
couple to the same Higgs boson. A counterexample was provided by Abe, Sato and
Yagyu in a muon-specific two-Higgs doublet model with a softly-broken $Z_4$
symmetry. In this model, all fermions except the muon couple to one Higgs and
the muon couples to the other. We study the phenomenology of the model and show
that there is a wide range of parameter-space in which the branching ratios of
the 125 GeV Higgs are very close to their Standard Model values, with the
exception of the branching ratio into muons, which can be substantially
suppressed -- this is an interesting possibility, since the current value of
this branching ratio is $0.5 \pm 0.7$ times the Standard Model value. We also
study the charged Higgs boson and show that, if it is lighter than $200$ GeV,
it could have a large branching ratio into $\mu\nu$ - even substantially larger
than the usual decay into $\tau\nu$. The decays of the heavy neutral scalars
are also studied. The model does have a relationship between the branching
ratios of the 125 GeV Higgs into $Z$'s, $\tau$'s and $\mu$'s, which can be
tested in future accelerators.
