Let's consider a free particle (I would come to the case of a particle under a non-trivial potential later). The eigenspace of such a particle corresponding to a particular energy $E$ is two-fold degenerate, in particular,
\begin{align}
\psi^+_{E}&=e^{-i(Et-\sqrt{2mE}x)/\hbar}\\
\psi^-_{E}&=e^{-i(Et+\sqrt{2mE}x)/\hbar}
\end{align}
where symbols have their usual meanings. As you can see, the wave-function is indeed a wave-solution with the (angular) frequency $\omega=E/\hbar$ and wave-number $k=\sqrt{2mE}/\hbar$. Thus, we get that the rough analog of the speed of light would be either the group velocity of the wave-function or the phase velocity of the wave-function. As you can calculate, the group velocity is $\frac{dw}{dk}=\frac{\hbar k}{m}=\sqrt{\frac{2E}{m}}$ and the phase velocity is $\frac{w}{k}=\frac{\hbar k}{2m}=\sqrt{\frac{E}{2m}}$.
Two points of caution:
- This framework cannot exactly handle the quantum mechanics of light because light is inherently relativistic and one needs to use quantum field theory to properly handle it.
- Finally, in the case of a particle under a non-trivial potential, the solutions are not wave-like and thus, the de Broglie principle is not valid. See: Dispersion relation of QM in the presence of a potential.
