Let $c=c(z, \bar z)$ be a complex function satisfying $\partial_{z} \bar c=\partial_{\bar z} c$. It follows that there exists a real function $f$ such that $\partial_{\bar z} f=-c$. Would it be possible to find a real function $g$ such that $\partial_{\bar z} g=-i c$ is satisfied where $i^2=-1$?

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If such a $g$ exists, then $f+ig$ is holomorphic, so $f$ is harmonic, so $\partial_z c=0$, i.e. $c$ is locally constant. But if $c$ is locally constant, then $f$ is harmonic, and so locally the real part of a holomorphic function, with imaginary part $g$, so $g$ exists locally.