I've been trying to understand contour integration from a more sophisticated perspective.I don't really like the fact that at a basic level one defines contour integration for $C^1$ curves using a parameterization. I am wondering if one can dispense with this notion and whether or not one even needs to consider the complex structure on the domain at all.
A $1$-dimensional current $T$ on $\mathbb{R}^2$ takes a smooth compactly supported $1$-form $\omega$ on $\mathbb{R}^2$ and gives a real number $T(\omega)$.
If I allow my $1$-form to be complex valued then of course I am allowed $dx + i dy$, which we can call "$dz$". It is clear how to pair this with a vector field. So now if I allow my currents to be complex linear, I get $$ T(dx + idy) = T(dx) + iT(dy) \in \mathbb{C}. $$ Is this a generalization of contour integration? So if $T$ is formed by multiplying a multiplicity 1 rectifiable current supported on $\gamma$ with an $\mathcal{H}^1$ measurable function $f : \mathbb{R}^2 \to \mathbb{C}$, then is it true that $T(dz) = \int_{\gamma} f(z) dz$ in the usual sense?
