  The complex Hilbert space of standard quantum mechanics may be treated as a
real Hilbert space. The pure states of the complex theory become mixed states
in the real formulation. It is then possible to generalize standard quantum
mechanics, keeping the same set of physical states, but admitting more general
observables. The standard time reversal operator involves complex conjugation,
in this sense it goes beyond the complex theory and may serve as an example to
motivate the generalization. Another example is unconventional canonical
quantization such that the harmonic oscillator of angular frequency $\omega$
has any given finite or infinite set of discrete energy eigenvalues, limited
below by $\hbar\omega/2$.
