  We study a two-dimensional exactly solvable non-Hermitian $PT-$non-symmetric
quantum model with real spectrum, which is not amenable to separation of
variables, by supersymmetrical methods. Here we focus attention on the property
of pseudo-Hermiticity, biorthogonal expansion and pseudo-metric operator. To
our knowledge this is the first time that pseudo-Hermiticity is realized
explicitly for a nontrivial two-dimensional case. It is shown that the
Hamiltonian of the model is not diagonalizable.
