  A new class of non-Hermitian Hamiltonians with real spectrum, which are
written as a real linear combination of su(2) generators in the form $ H=\omega
J_{3}+\alpha J_{-}+\beta J_{+}$, $\alpha \neq \beta$, is analyzed. The metrics
which allows the transition to the equivalent Hermitian Hamiltonian is
established. A pseudo-Hermitian supersymmetic extension of such Hamiltonians is
performed. They correspond to the pseudo-Hermitian supersymmetric systems of
the boson-phermion oscillators. We extend the supercoherent states formalism to
such supersymmetic systems via the pseudo-unitary supersymmetric displacement
operator method. The constructed family of these supercoherent states consists
of two dual subfamilies that form a bi-overcomplete and bi-normal system in the
boson-phermion Fock space. The states of each subfamily are eigenvectors of the
boson annihilation operator and of one of the two phermion lowering operators.
