  A comparative analysis of a model of complex scalar field $\phi$ and real
scalar field $\chi$ with interaction $g\phi^*\phi\chi$ for the real and purely
imaginary values of coupling $g$ in perturbative and non-perturbative regions.
In contrast to the usual Hermitian version (real $g$), which is asymptotically
free and energetically unstable, the non-Hermitian $PT$--symmetric theory
(imaginary $g$) is energetically stable and not asymptotically free. The
non-perturbative approach based on Schwinger--Dyson equations reveals new
interesting feature of the non-Hermitian model. While in the Hermitian version
of theory the phion propagator has the non-physical non-isolated singularity in
the Euclidean region of momenta, the non-Hermitian theory substantially free of
this drawback, as the singulatity moves in the pseudo-Euclidean region.
