In normal line integration, from what I understand, you are measuring the area underneath $f(x,y)$ along a curve in the $x\text{-}y$ plane from point $a$ to point $b$.
But what is being measured with complex line integration, when you go from a point $z_1$ to a point $z_2$ in the complex plane?
With regular line integration I can see $f(x,y)$ maps $(x,y)$ to a point on the $z$ axis directly above above/below $(x,y)$.
But in the complex case, when you map from the domain $Z$ to the image $W$, you are mapping from $\mathbb{R^2}$ to $\mathbb{R^2}$ ...it is not mapping a point to 'directly above/below'...so I don't have any intuition of what is happening with complex line integration.