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A harmonic oscillator Hamiltonian augmented by a non-Hermitian \pt-symmetricpart and its su(1,1) generalizations, for which a family of positive-definitemetric operators was recently constructed, are re-examined in a supersymmetriccontext. Quasi-Hermitian supersymmetric extensions of such Hamiltonians areproposed 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 Hamiltoniansrelated by similarity to Hermitian ones. Some examples of them are reviewed.

Quasi-Hermitian supersymmetric extensions of a non-Hermitian oscillator Hamiltonian and of its generalizations