  Motivated by the structure of the Swanson oscillator, which is a well-known
example of a non-hermitian quantum system consisting of a general
representation of a quadratic Hamiltonian, we propose a fermionic extension of
such a scheme which incorporates two fermionic oscillators, together with
bilinear-coupling terms that do not conserve particle number. We determine the
eigenvalues and eigenvectors, and expose the appearance of exceptional points
where two of the eigenstates coalesce with the corresponding eigenvectors
exhibiting the self-orthogonality relation. We compute the entanglement
spectrum and entanglement entropy of the ground state in two different ways,
with one of them being via the Gelfand-Naimark-Segal construction. In addition
to the approach involving the usual bi-normalization of the eigenvectors of the
non-hermitian Hamiltonian, we also discuss the case where the eigenvectors are
normalized with respect to the Dirac norms. It is found that the model exhibits
a quantum phase transition due to the presence of a ground-state crossing.
