  Supersymmetry offers one of the deepest insights in the concept of
solvability in quantum mechanics. This insight is, paradoxically, restricted by
one of the most serious formal drawbacks of the standard Witten's formulation
of supersymmetric quantum mechanics which lies in the Jevicki-Rodrigues'
postulate of absence of poles in superpotentials W(x) over all the real axis of
coordinates x. In our review we emphasize that this obstacle is artificial and
that it disappears immediately after a suitable (say, constant) shift of the
axis of x into complex plane. Detailed attention is paid to a close
relationship between this common trick and the recent not quite expected
increase of interest in non-Hermitian (a. k. a. PT-symmetric or
pseudo-Hermitian) Hamiltonians. We show that the resulting PT-SUSY
regularization recipe proves both easy and universal. An insight into its
mathematics is mediated by the complex harmonic oscillator with a
centrifugal-like spike. An exhaustive discussion of the role of the strength of
this spike is offered. In addition we recollect the possibility of a
re-formulation of the recipe in the second-order SUSY language. Finally we list
a few promising directions of applicability of our PT-SUSY regularization
prescription to a few more complicated nonrelativistic models (superintegrable
Hamiltonians of the Smorodinsky-Winternitz and of the Calogero-Sutherland type)
and to the relativistic Klein-Gordon equation (as well as to all of its
unphysical higher-order analogues).
