Mathematical significance of the "Dirac conjugate" 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.)

The Lorentz group, $SO(1,3)$, is non-compact, thus their representations are not unitary (in general).
Therefore, if you have a spinor, $\psi\in \mathcal{S}$, transforming as $\psi\mapsto S\psi$, it follows that the construction $$ \psi^\dagger \psi \mapsto \psi^\dagger S^\dagger S \psi \neq\psi^\dagger \psi,$$
since $S^\dagger\neq S^{-1}$.
This tells us that $\psi^\dagger$ does not belongs to the dual space of the spinors, $\psi^\dagger\not\in \mathcal{S}^*$.
It this point, you can realize that $$S^\dagger\gamma^0 = \gamma^0 S^{-1},$$
and this allows to define a dual spinor to $\psi$ through the construction $$\mathcal{S}^*\ni\bar{\psi}\equiv \psi^\dagger\gamma^0.$$
Hope it would be helpful.
Summary
The Dirac conjugate serves to define a dual spinor, by giving a spinor in the direct space.
