  The issue of Hermiticity of the Higgs boson interaction with fermions is
addressed. A model for non-Hermitian Yukawa interaction is proposed and
approximation of one fermion generation is considered. Symmetry properties of
the corresponding $h f \bar{f}$ Lagrangian with respect to the discrete P, C
and T transformations are analyzed, and the modified Dirac equation for the
free fermion is studied. Longitudinal polarization of the fermions in the decay
$h \to f \bar{f}$, which arises due to non-Hermiticity of the $h f \bar{f}$
interaction, is discussed. It is suggested to study effects of this
non-Hermiticity in the decay $h \to \tau^- \tau^+ \to \mu^- {\bar \nu}_\mu
\nu_\tau \, \mu^+ \nu_\mu {\bar \nu}_\tau$, for which observables (asymmetries)
are constructed which take nonzero values for a non-Hermitian $h \tau^- \tau^+$
interaction. These asymmetries are analyzed for various configurations of the
muon energies.
