I study physics so apologies for any nonstandard notation/terminology.
In geometric (aka spacetime) algebra one speaks of basis $\gamma^{\mu}$ ( possibly represented by matrices) as transforming in a two sided way.
$$\bar{\gamma^{\mu}}=\psi\gamma^{\mu}\psi^{-1}$$
For a general pseudo-Riemannian metric suppose we have the following rather standard relation:
$$g^{\mu\nu}I_{4}=\frac{1}{2}\left(\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}\right)$$
It's evident then that:
$$\bar{g^{\mu\nu}}=\psi g^{\mu\nu}\psi^{-1}$$
If however, we are considering a wider class of transformations such that $\psi$ is not orthogonal (ie. $\psi^{\dagger}\neq\psi^{-1}$ ), where the dagger denotes the conjugate transpose. one has that:
$$\bar{\gamma^{\mu}}=\psi\gamma^{\mu}\psi^{\dagger}$$
And therefore:
$$\bar{g^{\mu\nu}}I_{4}=\frac{1}{2}\psi\left(\gamma^{\mu}\psi\psi^{\dagger}\gamma^{\nu}+\psi\gamma^{\nu}\psi\psi^{\dagger}\gamma^{\mu}\right)\psi^{\dagger}$$
Is this then the correct transformation for the metric? What, if anything, can we say about the quantity $\psi\psi^{\dagger}$ (other than it being hermitian of course)?