  A method to construct non-Dirac-hermitian supersymmetric quantum system that
is isospectral with a Dirac-hermitian Hamiltonian is presented. The general
technique involves a realization of the basic canonical (anti-)commutation
relations involving both bosonic and fermionic degrees of freedom in terms of
non-Dirac-hermitian operators which are hermitian in a Hilbert space that is
endowed with a pre-determined positive-definite metric. A pseudo-hermitian
realization of the Clifford algebra for a pre-determined positive-definite
metric is used to construct supersymmetric systems with one or many degrees of
freedom. It is shown that exactly solvable non-Dirac-hermitian supersymmetric
quantum systems can be constructed corresponding to each exactly solvable
Dirac-hermitian system. Examples of non-Dirac-hermitian (i) non-relativistic
Pauli Hamiltonian, (ii) super-conformal quantum system and (iii) supersymmetric
Calogero-type models admitting entirely real spectra are presented.
