Let $\Gamma$ be a smooth curve in $\mathbb{C}^d$. Since $\mathbb{C}^d$ can be seen as $\mathbb{R}^{2d}$, one can define the line integral of functions $f:\Gamma\to \mathbb{C}$ using for instance arclength parametrization.

However, when $d=1$, one can use the usual complex line integral of complex analysis instead. Is there a natural extension of this definition when $d \geq 2$ ?