We present the theory of the complex Korteweg-de Vries (KdV) equation in the hydrodynamics of shallow fluids. Complex KdV theory is a substantial generalisation of real KdV theory, which we show reduces to real KdV theory as a special case. However, it also allows us to easily describe fluid trajectories beneath the surface without any numerical simulation. We are also able to give a more general interpretation to well known conservation laws, as well as give a physical meaning to the Ablowitz-Kaup-Newell-Segur (AKNS) system. Finally, we also present some results on newly discovered symmetries of KdV rational solutions.

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