  A harmonic oscillator Hamiltonian augmented by a non-Hermitian \pt-symmetric
part and its su(1,1) generalizations, for which a family of positive-definite
metric operators was recently constructed, are re-examined in a supersymmetric
context. Quasi-Hermitian supersymmetric extensions of such Hamiltonians are
proposed by enlarging su(1,1) to a ${\rm su}(1,1/1) \sim {\rm osp}(2/2, \R)$
superalgebra. This allows the construction of new non-Hermitian Hamiltonians
related by similarity to Hermitian ones. Some examples of them are reviewed.
