How to generalize the momentum reversal operator?

Question

In non-relativistic quantum mechanics, in position space, the complex conjugate operator $C$ flips the sign of the momentum operator, $CpC=-p$ (and thus also flips the sign of the orbital angular momentum).
The operator $T=C$ is usually called the time reversal operator, but this only works in certain cases. For example, it does not work in the presence of spins  (where $T=Cisigma_y$). So $C$ may not always be called the (full) time reversal operator.
Would it be safe to say that $C$ is the momentum reversal operator? I think this holds always. Also, does $C$ do anything else that is not associated to momentum directly? Is the general behavior of $C$ well-studied? References would be much appreciated.
Edit: after writing this, I realized that it also flips the sign of $ipartial_t$ (some kind of energy reversal?), but I am not sure how helpful it can be to talk of non-stationary states (or of relativistic quantum mechanics).
Edit: I now realize that this question is ill defined, I did not want people to focus on the basis issue. A better question would be to ask specifically how to generalize the $C$ operator to other basis in relation to time and momentum.

mike stone · Answer

The "complex conjugation" operator is basis dependent and so ill defined.
A vector  that has real components in one basis may be have  complex components in another. For example the $xleftrightarrow p$ change-of-basis formula for the components is
$$
tilde psi(p) = langle p|psirangle = int dx langle p|xranglelangle x|psirangle= int dx e^{ipx}psi(x)
$$
and the $i$ in $e^{-ipx}$ means that a state $|psirangle$ that has real components $psi(x)=langle x|psirangle$ in the $|xrangle$ basis can have complex components  $tilde psi(p)=langle p|psirangle$ in the $|prangle$ basis.
Correspondingly
the $hat p$ opertor acts  as  $-ipartial _x$ in the $|xrangle$  basis and so flips sign under congugation, while $hat x$ which acts by multiplication by the real number $x$ does not. But in the $|prangle$ basis $hat p$ is just multiplication by the real number $p$ while $hat x$ acts as $ipartial_p$, so the "flips" are reversed.
As consequence of this, "complex conjugation" can only be defined as an operator if you also specify a set of basis vectors that are declared "real"
Thus, when defining antilinear operators such as time reversal it is dangerous  to  factor the operator as "$T=Cisigma_y$" -- although this is common practice. It is confusing because this formula  is only correct when acting on states in the $sigma_x$-diagonal or $sigma_x$-diagonal bases which  are connected by change-of-basis matrices with real entries. If you have need to use a different spin basis, say the general ${bf n}cdot {boldsymbol sigma}$-diagonal basis, the formula for $T$ is different.

How to generalize the momentum reversal operator?

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