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][1] 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 an electron which would be not only relativistic and implied the Klain-Gordon equation, but also $|\psi|^2$ would be the time component of a conserved 4-current.


  [1]: https://en.wikipedia.org/wiki/Skolem%E2%80%93Noether_theorem