How do I visualise the contour integral of a complex function?

Question

I've just learnt about the contour integral of a complex function, but I'm having trouble figuring out what it is calculating visually. I understand it is somewhat analogous to the line integral for real functions so should I be thinking about it like the $\mathbb{R}^2 \rightarrow \mathbb{R} $ version of that (as the complex plane is isomorphic to $ \mathbb{R}^2)? $