What about a Hermitian metric with an imaginary EM potential?

$g_{ab} = \left[\matrix{ -1 & -i A_j \\ i A_i & \delta_{ij} }\right]$

$g_{tt} = -1$
$g_{it} = \bar{g}_{ti} = i A_i$
$g_{ij} = \delta_{ij}$

$g_{ab} = \eta_{ab} + 2 i A_{[a} \delta_{b]t} $

Well right off the bat, if we choose an imaginary EM potential and Hermitian metric
then we must make the imaginary portion of $g_{tt}$ antisymmetric, so we lose $A_t$.
This makes the electric field difficult to derive.