WireWorld is one of the more interesting cellular automata I've encountered. It tries to simulate a simple model of an electrical process, which uses only four states. Even with this simple model, directional flows can be established, and logic circuits can be constructed. I show it here with an electron head/tail pair traveling around the outer edge, with the inner area initialized with noise. Sometimes you can see some very elaborate behavior from the CA operating inside of these noise patterns.

The four states are: empty(grey), conductor(black), electron head(yellow), and electron tail(cyan). The update rules are simple. Empty becomes emtpy. Electron head becomes electron tail. Electron tail becomes conductor. A conductor with exactly one or two electron heads in its neighborhood becomes an electron head.

People have constructed functioning computers out of digital logic constructed with this CA. For example, this is an example of a decoding circuit to turn a serial signal into a control signal for the seven segment display. I have also seen logic gates, diodes, oscillators of a tunable period, and more, even an implemenation of an instruction set capable of performing a basic calculation. It is a very wild example of a turing complete automata.