Complex integral of a vector field is zero

Question

Let $\Omega \subset \mathbb{R}^2$ be an open domain with smooth boundary. Identifying $\mathbb{R}^2$ with $\mathbb{C},$ consider the following integral $$ \int_{\Omega} \Big(f(z)\frac{z}{|z|} - g(z)\frac{\bar z}{|z|} \Big) dz \wedge d\bar z =0. $$ Note that the unit vectors $\frac{z}{|z|}$ and $- \frac{\bar z}{|z|}$ are orthogonal to each other. What can we say about the vector field $(f(z),g(z))?$ Can we get any information on its 2D-curl or divergence?