  In recent work Alexandre, Ellis, Millington and Seynaeve have extended the
Goldstone theorem to non-Hermitian Hamiltonians that possess a discrete
antilinear symmetry such as $PT$. They restricted their discussion to those
realizations of antilinear symmetry in which all the energy eigenvalues of the
Hamiltonian are real. Here we extend the discussion to the two other
realizations possible with antilinear symmetry, namely energies in complex
conjugate pairs or Jordan-block Hamiltonians that are not diagonalizable at
all. In particular, we show that under certain circumstances it is possible for
the Goldstone boson mode itself to be one of the zero-norm states that are
characteristic of Jordan-block Hamiltonians. While we discuss the same model as
Alexandre et al. our treatment is quite different, though their main conclusion
that one can have Goldstone bosons in the non-Hermitian case remains intact. In
their paper Alexandre et al. presented a variational procedure for the action
in which the surface term played an explicit role, to thus suggest that one has
to use such a procedure in order to establish the Goldstone theorem in the
non-Hermitian case. However, by taking certain fields that they took to be
Hermitian to actually either be anti-Hermitian or be made so by a similarity
transformation, we show that we are then able to obtain a Goldstone boson using
a completely standard variational procedure. Since we use a standard
variational procedure we can readily extend our analysis to a continuous local
symmetry by introducing a gauge boson. We show that when we do this the gauge
boson acquires a non-zero mass by the Higgs mechanism in all realizations of
the antilinear symmetry, except the one where the Goldstone boson itself has
zero norm, in which case, and despite the fact that the continuous local
symmetry has been spontaneously broken, the gauge boson remains massless.
