  It is known that the anti-Wick (or standard coherent state) quantization of
the complex plane produces both canonical commutation rule and quantum spectrum
of the harmonic oscillator (up to the addition of a constant). In the present
work, we show that these two issues are not necessarily coupled: there exists a
family of separable Hilbert spaces, including the usual Fock-Bargmann space,
and in each element in this family there exists an overcomplete set of
unit-norm states resolving the unity. With the exception of the Fock-Bargmann
case, they all produce non-canonical commutation relation whereas the quantum
spectrum of the harmonic oscillator remains the same up to the addition of a
constant. The statistical aspects of these non-equivalent coherent states
quantizations are investigated. We also explore the localization aspects in the
real line yielded by similar quantizations based on real Hermite polynomials.
