  We consider the problem of designing a variety of "system guided" basis sets
for quantum mechanical anharmonic oscillators. Using ideas based on
supersymmetric quantum mechanics, we design canonical transformations of the
usual position and momentum to generate generalized "Cartesian-like positions,
W and momenta, p_W" with unit Poisson brackets. These are quantized following
Dirac, leading to an infinite family of potential "operator observables". The
fundamental issue is that all but one of the operators are not Hermitian
(formally self-adjoint) in the original position representation. We show that
the resulting operators are "quasi-Hermitian" relative to the x-representation
and that all are Hermitian in the W-representation. Depending on how one treats
the Jacobian of the canonical transformation in the expression for the
classical momentum, p_W, quantization yields a) continuous mutually unbiased
bases (MUB) b) orthogonal bases (with Dirac delta normalization) c)
biorthogonal bases (with Dirac delta normalization) d) new W-harmonic
oscillators yielding standard orthonormal bases (as functions of W) and
associated coherent states. The MUB include W-generalized Fourier transform
kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the
spectrum (p/m i, p/m i). The W, p_W satisfy the uncertainty product relation:
Delta-W * Delta-p_W gte 1/2.
