The context is deriving cauchy riemann equations using green's/stoke's theorem. The function is the complex function $f(x,y)=u(x,y)+iv(x,y)$ with associated one form $u(x,y)dx+iv(x,y)dy$.
Here is my work so far:
$$d(u(x,y)dx+iv(x,y)dy)=(\frac{\partial u(x,y)}{\partial x}dx +\frac{\partial u(x,y)}{\partial y}dy)\wedge dx+i(\frac{\partial v(x,y)}{\partial x}dx +\frac{\partial v(x,y)}{\partial y}dy)\wedge dy=(-\frac{\partial u(x,y)}{\partial y}+ i\frac{\partial v(x,y)}{\partial x})dx\wedge dy$$
Is this correct? Should there be an $i$ coefficient on the differential $dy$ since it is in the purely imaginary direction?
I haven't been able to find many resources on complex exterior differentiation, so any help is appreciated.
The exterior derivative operator $d$ is real (i.e., real-valued on real-valued forms) and complex-linear (i.e., if $\alpha$ and $\beta$ are real $p$-forms, then $d(\alpha + i\beta) = d\alpha + i\, d\beta$, so yes, your calculation is correct.
To deduce the Cauchy-Riemann equations from Green's theorem, it's convenient to differentiate the holomorphic $1$-form $$ f(z)\, dz = (u + iv)(dx + i\, dy). $$