Let $\psi$ be a Dirac spinor. The so-called "Dirac conjugate" of $\psi$ is defined to be $\widetilde{\psi}:=\psi ^*\gamma ^0$, where $^*$ denotes the adjoint and the gamma matrices $\gamma ^\mu$ comprise the essentially unique irreducible representation of $\mathcal{C}\ell (1,3)$. Physicists introduce this, in a relatively ad hoc manner, so that the quantity $$ \widetilde{\psi}\psi $$ is Lorentz invariant. This does the trick, but I have a feeling like there is something deeper going on here.
The quantity $\psi ^*\gamma ^0$ makes sense for an arbitrary Clifford algebra $\mathcal{C}\ell (1,2m-1)$, whereas the notion of Lorentz invariance is specific to the case $m=2$, so the significance of $\widetilde{\psi}$ in other dimensions is not obvious to me. It might be the case that the significance of $\psi ^*\gamma ^0$ is unique to the case $m=2$, but I would be surprised if that were the case.
So then, what is the general mathematical significance of the Dirac conjugate $\psi ^*\gamma ^0$.
(Please let me know if explanation of any physics jargon is needed.)