the real and imaginary parts are harmonic
Here is one of the first surprising payoffs of holomorphy. If f = u + i v is holomorphic, then each of its real and imaginary parts, taken on its own as a function of (x, y), automatically satisfies Laplace's equation: u_xx + u_yy = 0 and v_xx + v_yy = 0. We say u and v are harmonic. So every holomorphic function quietly produces two solutions of the most important partial differential equation in physics, for free.
The derivation is a short calculation from the Cauchy-Riemann equations, assuming the partials are continuous (true for holomorphic functions, which are in fact infinitely differentiable). From u_x = v_y differentiate in x: u_xx = v_yx. From u_y = -v_x differentiate in y: u_yy = -v_xy. Adding, u_xx + u_yy = v_yx - v_xy = 0, because mixed partials are equal. The same trick gives v_xx + v_yy = 0.
This bridge runs in both directions and is hugely useful. It means the real part of any holomorphic function — temperature in steady-state heat flow, electric potential in electrostatics, velocity potential in ideal fluid flow — is harmonic, so complex analysis becomes a powerful tool for two-dimensional potential problems. Conversely, given a harmonic u on a nice (simply connected) region, you can find a harmonic conjugate v and recover a holomorphic f = u + i v, turning a hard PDE into the algebra of holomorphic functions.
From f(z) = z^2, u = x^2 - y^2: u_xx = 2, u_yy = -2, so u_xx + u_yy = 0 — harmonic. Likewise v = 2xy has v_xx = 0, v_yy = 0, so v_xx + v_yy = 0. Both parts solve Laplace's equation.
Differentiating the Cauchy-Riemann equations and using equality of mixed partials proves both parts harmonic.
The argument quietly uses that the partials are smooth enough for mixed partials to commute — fine for holomorphic functions, which are infinitely differentiable, but it is an assumption being relied on.