  We extend the study of supersymmetric tridiagonal Hamiltonians to the case of
non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find
the relation between matrix elements of the non-Hermitian Hamiltonian $H$ and
its supersymmetric partner $H^{+}$ in a given basis. Moreover, the orthogonal
polynomials in the eigenstate expansion problem attached to $H^{+}$ can be
recovered from those polynomials arising from the same problem for $H$ with the
help of kernel polynomials. Besides its generality, the developed formalism in
this work is a natural home for using the numerically powerful Gauss quadrature
techniques in probing the nature of some physical quantities such as the energy
spectrum of $\mathcal{PT}$-symmetric complex potentials. Finally, we solve the
shifted $\mathcal{PT}$-symmetric Morse oscillator exactly in the tridiagonal
representation.
