  We examine the properties and consequences of pseudo-supersymmetry for
quantum systems admitting a pseudo-Hermitian Hamiltonian. We explore the Witten
index of pseudo-supersymmetry and show that every pair of diagonalizable (not
necessarily Hermitian) Hamiltonians with discrete spectra and real or
complex-conjugate pairs of eigenvalues are isospectral and have identical
degeneracy structure except perhaps for the zero eigenvalue if and only if they
are pseudo-supersymmetric partners. This implies that pseudo-supersymmetry is
the basic framework for generating non-Hermitian PT-symmetric and
non-PT-symmetric Hamiltonians with a real spectrum via a Darboux
transformation, and shows that every diagonalizable Hamiltonian H with a
discrete spectrum and real or complex-conjugate pairs of eigenvalues may be
factored as H=L^# L where L is a linear operator with pseudo-adjoint L^#. In
particular, this factorization applies to PT-symmetric and Hermitian
Hamiltonians. The nondegenerate two-level systems provide a class of
Hamiltonians that are pseudo-Hermitian. We demonstrate the implications of our
general results for this class in some detail.
