What are functions of a complex variable used for in physics?
Complex analysis is more than just a tool that can be used for computing difficult integrals. For example:
In quantum field theory, one of the most popular regularization schemes relies on the theory of complex functions. In particular, it relies on the concept of analytic continuation of functions $f:D\to\mathbb C$ for some $D\subseteq \mathbb C$. In particular, one often encounters integrals that are divergent in four dimensions, so one first computes the same integral in arbitrary positive integer dimension $d$, and then one regards this expression as a function of a complex variable $d$ and analytically continues it to a meromorphic function on the complex plane. This allows one to then parameterize the divergence that appears in the integral in the parameter $\epsilon = d-4$, conduct renormalization, and then take $\epsilon\to 0$. This is a case in which complex analysis isn't being used just as a trick, it's being used to render the expressions in perturbative quantum field theory well-defined. The procedure outlined above is called dimensional regularization.
In the context of two-dimensional conformal field theory (CFT), the fields of the theory can be regarded as functions on the complex plane. If you open any CFT text, it'll basically just seem like an advanced complex analysis textbook.
You mention Laplace's equation (which comes up in lots of physical contexts like electrostatics, heat flow, fluid flow, etc.), but here's another facet you didn't mention. Laplace's equation is invariant under conformal transformations which are themselves mappings $\mathbb C\to \mathbb C$, an the theory of such functions can be used to solve Laplace's equation boundary value problems.
In quantum field theory and string theory, zeta regularization can be used to give meaning to divergent sums by using the Riemann zeta function which is a function of a complex variable. For example, some derivations of the Casimir effect use zeta regularization.
This is all very fine, but I've never seen anyone talk about functions of a complex variable.
The Laplace transform is a widely used integral transform in mathematics with many applications in physics and engineering. It is a linear operator of a function f(t) with a real argument t (t ≥ 0) that transforms f(t) to a function F(s) with complex argument s, given by the integral
$$F(s) = \mathcal L\{f(t)\}= \int_0^\infty f(t)e^{-st}dt, \, s = \sigma +i\omega $$
Since, by the above
$$\mathcal L\{\frac{d}{dt}f(t)\} = sF(s) - f(0^-)$$
and
$$\mathcal L\{\int_0^tf(\tau)d\tau\} = \frac{1}{s}F(s)$$
the Laplace transform of a differential or integral equation is an algebraic equation in the complex variable $s$.
An elementary application is to the Tautochrone problem.
Integration over complex contours is also crucial to one of the main methods of obtaining asymptotic approximations for a large class of special functions and theoretical results, the stationary phase approximation and the method of steepest descent.
Those methods are a huge aid in calculating integrals like $$ \int_0^\pi \cos(x\cos(\theta))\text d\theta \quad \text{when }x\gg1, $$ which are hard to calculate, even numerically, because the integrand is highly oscillatory and you need to deal with a lot of cancellations. The idea is to express the integral as a contour integral, and then change the contour so that the integrand will look like a gaussian with a flat phase, which is easy to integrate. This sort of method is the backbone of most of the special function asymptotics, e.g. those for Bessel, Airy, Gamma or hypergeometric functions, among many others.
This sort of analysis is crucial in directly analysing many physical problems, and it is certainly my own daily bread and butter. To give some examples:
It permits a very intuitive treatment of non-adiabatic transitions for a two-level system in the presence of a not-quite-adiabaticly varying perturbation, in the form of the Landau-Zener formula (doi).
It is the driving force in any simple treatment of tunnelling and particularly tunnelling ionization. The main analytical and conceptual tool for dealing with High Harmonic Generation, for example, the strong-field approximation, is built using these tools.
The method of stationary phase is behind the passage from quantum theory to classical mechanics in the form of the principle of stationary phase, as in e.g. this SE question.
Similarly, it plays a large role in optics and particularly in describing caustics.
In fact, this part of mathematical physics is growing so rapidly that a part of the method is one of the newest additions to the Digital Library of Mathematical Functions (also known as the NIST Handbook of Mathematical Functions, the successor of Abramowitz and Stegun).