Let $(\eta^{\mu\nu})=diag(+1,-1,-,1-,1)$ be the diagonal matrix. The Dirac matrices $\gamma^\mu, \mu=0,1,2,3$ satisfy by definition $$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu},\,\,\,(1)$$ 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 (1) then the Skolem-Noether theorem implies that there exists an invertible matrix $C$ such that $$\tilde\gamma^\mu=C\gamma^\mu C^{-1}\mbox{ for all } \mu.\,\,\,(2)$$

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

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

Remark 2. My understanding is that the motivation to consider conditions (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 $|\psi|^2$ would be the time component of a conserved 4-current.