the complex potential
Steady physical fields in the plane often come in pairs: a temperature and the lines of heat flow, a voltage and the lines of electric force, a flow speed and the streamlines a particle would follow. The complex potential is the single holomorphic function that packages both members of such a pair into one object, so the whole problem can be handled with the tools of complex analysis at once.
Start with a harmonic function phi (a potential — say the velocity potential of a smooth, swirl-free, incompressible fluid flow). Because phi is harmonic, on a nice domain it has a harmonic conjugate psi, and together they form a holomorphic function Omega(z) = phi(x, y) + i psi(x, y), the complex potential. The two real parts have crisp physical meanings: the level curves phi = constant are the equipotential lines (perpendicular to the flow), and the level curves psi = constant are the streamlines (the actual paths the fluid follows); psi is called the stream function. Better still, the flow velocity is read off by a single derivative: the complex velocity is the conjugate of Omega'(z), so Omega'(z) = u - i v packages both velocity components, and where Omega'(z) = 0 you have a stagnation point. The Cauchy-Riemann equations relating phi and psi are exactly the physical statements that the flow is irrotational and divergence-free.
The power of the idea is that conformal maps move solutions around. Because Omega is holomorphic, composing it with a conformal map sends a known flow on a simple region (say uniform flow past a flat plate, or flow in a half-plane) to the flow on a complicated region (flow past an airfoil), with equipotentials and streamlines carried along automatically — this is how the Joukowski airfoil and the lift it generates are computed. The honest scope: the complex-potential picture is special to two dimensions and to fields that are both irrotational and source-free (so the potential is harmonic); genuinely three-dimensional flows, or flows with vorticity or sources, do not collapse into a single holomorphic function this way.
Uniform flow with speed U in the x-direction has complex potential Omega(z) = U z = U x + i U y, so phi = U x (equipotentials are vertical lines) and the stream function psi = U y (streamlines are horizontal lines); the velocity is Omega'(z) = U, a constant rightward flow.
The simplest complex potential Omega = U z describes uniform horizontal flow, streamlines psi = U y.
The complex potential exists as a single-valued function only when the harmonic conjugate exists globally — on a multiply-connected region (like flow around an obstacle with circulation) Omega can be multivalued, its imaginary part jumping by the circulation each time you loop the obstacle; that jump is exactly what produces lift.