The fact that a complex analytic function is made of a pair of real harmonic functions is well known. Therefore, I have the feeling that Cauchy-Goursat theorem, which practically means "the value of $f$ inside a domain is dictated by its boundary values" can be derived through an integral representation of the solution to Laplace equation.
Somewhat related, Morera theorem claims that if $f$ is continuous over a simply connected domain and its integral over any simple closed curve vanishes, than it's analytic. It sounds like one can express $f$ as a vector field and derive this results from the field being conservative.
I believe these two originate from the same thing, as a vector field generated as the grad of an harmonic function is conservative.
Other examples: Maximum Moduli Theorem, the fact that an analytic function which is constant in a region is constant everywhere.
In both cases I'm a bit confuse, as the complex line integral is something between the real line integral of a scalar field (1st type) to that of a vector field (2nd type), and I didn't find a way to express one as the other.