  I study indirect CP violation for neutral kaons, and extend it to large
values of the CP-violating parameter (taken to be real). I show how and at
which condition there can exist a continuous set of basis in which the kinetic
and mass terms in the Lagrangian can be diagonalized simultaneously. An
ambiguity results for the mass spectrum, which then depends on the basis. In
particular, for fixed (positive) (mass)^2 of the CP eigenstates K^0_1, K^0_2,
and for certain ranges of values of the CP-violating parameter, a negative
(mass)^2 can arise in the CP-violating basis. Under certain conditions, even a
small perturbation, by lifting the ambiguity, can strongly alter the pattern of
masses. These investigations extend in a natural way to indirect CP violation
among a set of Higgs-like doublets. The C-odd commutator [K^0, K^0 bar], or its
equivalent for Higgs multiplets, plays an important role. The condition for its
vanishing and its consequences are among the main concerns of this work.
