  We demonstrate that the recent paper by Abhinav and Panigrahi entitled
`Supersymmetry, PT-symmetry and spectral bifurcation' [Ann.\ Phys.\ 325 (2010)
1198], which considers two different types of superpotentials for the
PT-symmetric complexified Scarf II potential, fails to take into account the
invariance under the exchange of its coupling parameters. As a result, they
miss the important point that for unbroken PT-symmetry this potential indeed
has two series of real energy eigenvalues, to which one can associate two
different superpotentials. This fact was first pointed out by the present
authors during the study of complex potentials having a complex $sl(2)$
potential algebra.
