Let $(\eta^{\mu\nu})=\operatorname{diag}(+1,-1,-1,-1)$. The Dirac matrices $\gamma^\mu$, $\mu=0,1,2,3$ satisfy by definition $$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\tag{1}\label{1}$$ where $\{A,B\}=AB+BA$ is the anti-commutator.

If we have another collection $\tilde\gamma^\mu$ of matrices of the same size satisfying the same commutation relations \eqref{1} then the Skolem–Noether theorem implies that there exists an invertible matrix $C$ such that $$\tilde\gamma^\mu=C\gamma^\mu C^{-1}\mbox{ for all } \mu.\tag{2}\label{2}$$

I am interested in the following situation. Let us require in addition to \eqref{1} that for Hermitian conjugates one has $$(\gamma^0)^*=\gamma^0, (\gamma^i)^*=-\gamma^i \text{ for }i=1,2,3.\tag{3}\label{3}$$ Is it possible to classify four tuples of matrices $\gamma^\mu$, $\mu=0,\dotsc,3$, satisfying \eqref{1} and \eqref{3}? Is there a generalization of such a classification to higher dimensions?

Remark 1. If one takes in \eqref{2} the matrix $C$ to be unitary then $\tilde\gamma^\mu$ also satisfy \eqref{3}.

Remark 2. My understanding is that the motivation to consider conditions \eqref{2} goes back to the original motivation of Dirac to construct an equation on the wave function $\psi$ of a free electron which would be not only relativistic and implied the Klein–Gordon equation, but also $\lvert\psi\rvert^2$ would be the time component of a conserved 4-current.