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 $h to tau^- tau^+ to mu^- {bar u}_mu u_tau , mu^+ u_mu {bar u}_tau$, 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.
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