  Hitherto, it is well known that complex PT-symmetric Scarf II has real
discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal
new interesting complex, non-PT-symmetric parametric domains of this versatile
potential, $V(x)$, where the spectrum is again discrete and real. Showing that
the Hamiltonian, $p^2/2m+V(x)$, in the new cases is pseudo-Hermitian could be
challenging, if possible.
