There isn't necessarily a right answer, but contour integration certainly has a rich theory that essentially lays the foundation for all elementary complex analysis. The Cauchy Integral Theorem/Formula is a pretty huge deal, and also allows us to extend the theory to things like the holomorphic functional calculus. Interestingly enough, the more general Cauchy Integral Formula for a $C^1$ function $f:\mathbb{C}\to\mathbb{C}$ actually involves both contour integrals and area integrals:
$f(z) = \frac1{2\pi i} \left(\oint_{\partial \Omega}\frac{f(w)}{w-z}dw - \iint_{\Omega}\frac{\partial f}{\partial \overline{w}}\frac1{w-z}d\overline{w}dw\right)$
where the boundary $\partial \Omega$ is piecewise $C^1$. If you're interested in this, I would suggest this article by Steven Krantz.
For a more natural interpretation of the contour integral, Polya associated holomorphic function $f(z)$ with the vector field
$W(z) = \left[Re\left(\overline{f(z)}\right),Im\left(\overline{f(z)}\right)\right] = \left[Re\left(f(z)\right),-Im\left(f(z)\right)\right]$
which is both irrotational (having zero two-dimensional curl) and incompressible (having zero divergent) due to the Cauchy-Riemann equations. Given a path $\Gamma$ in the complex plane, the real part of the contour integral $\int_\Gamma f(w) \ dw$ can be interpreted as the work done by $W$ along $\Gamma$, and the imaginary part as the flux. Tristan Needham gives a fantastic exposition of Polya vectors fields in the final chapters of his book Visual Complex Analysis, and Polya's own Complex Variables makes connections with fluid flows throughout.
The big location where I've seen complex area integrals is the Bergman space. The typical (Hilbert) Bergman space on the disc $\mathbb{D} = \{z \in \mathbb{C} : |z|<1\}$ is the collection of holomorphic functions $f:\mathbb{D}\to\mathbb{C}$ such that
$\iint_\mathbb{D} |f(w)|^2 dA(w) <\infty$
The collection of such functions forms a Hilbert space with inner product given by
$\langle f,g \rangle = \left(\iint_\mathbb{D} f(w)\overline{g(w)}dA(w)\right)^{1/2}$
One object of interest in the Bergman space is a function called the reproducing kernel. On $\mathbb{D}$, the reproducing kernel is given by
$K(z,w) = \frac1\pi \frac1{(1-z\overline{w})^2}$
$K(z,w)$ thought of as a function of $w$ can be used to pull out the $z$-value of functions on the Bergman space, as it has the property that
$f(z) = \iint_\mathbb{D} K(z,w)f(w)dA(w)$
for functions $f$ in the Bergman space. You can also look at Bergman spaces on domains other than the disc $\mathbb{D}$, which have their own reproducing kernels. A good introduction to the Bergman space is Krantz's Geometric Function Theory.