  An extension of the scope of quantum theory is proposed in a way inspired by
the recent heuristic as well as phenomenological success of the use of
non-Hermitian Hamiltonians which are merely required self-adjoint in a Krein
space with an indefinite metric (chosen, usually, as the operator of parity).
In nuce, the parity-like operators are admitted to represent the mere
indefinite metric in a Pontryagin space. A constructive version of such a
generalized quantization strategy is outlined and, via a non-numerical
illustrative example, found feasible.
