  The $\cal PT$-symmetric complexified Scarf II potential $V(x)= - V_1
\sech^{2}x + {\rm i} V_2 \sech x \tanh x$, $V_1>0$ , $V_{2}\neq 0$ is revisited
to study the interplay among its coupling parameters. The existence of an
isolated real and positive energy level that has been recently identified as a
spectral singularity or zero-width resonance is here demonstrated through the
behaviour of the corresponding wavefunctions and some property of the
associated pseudo-norms is pointed out. We also construct four different
rationally-extended supersymmetric partners to $V(x)$, which are $\cal
PT$-symmetric or complex non-$\cal PT$-symmetric according to the coupling
parameters range. A detailed study of one of these partners reveals that SUSY
preserves the $V(x)$ spectral singularity existence.
