The space-time diagram produced by this cellular automaton is indeed a great example of complex mathematical object (such as fractals), appearing from simple rules. I would suggest the following five ideas (only the two last are cellular automata):
Start with an equilateral triangle. For each side: Divide it into three equal parts. Replace the middle part with two sides of a smaller equilateral triangle pointing outward (a “bump”). Repeat the process recursively for each new segment. The limiting figure is the following fractal:
Take a strip of paper. Fold it in half always in the same direction (e.g., right over left). Unfold the strip so that each crease makes a right angle. After $n$ folds, the pattern of left and right turns forms a complex, self-similar curve known as the dragon curve.
Same idea for the Lévy C curve and other Lindenmayer system
Only using $1$s and $2$s, write a sequence that is exactly the sequence of length of its blocks. There is only one such sequence starting with $1$: $$K=\underbrace{1}_{\displaystyle 1}\,,\underbrace{2\,,2}_{\displaystyle 2},\underbrace{1\,,1}_{\displaystyle 2},\underbrace{2}_{\displaystyle 1},\underbrace{1}_{\displaystyle 1},\underbrace{2\,,2}_{\displaystyle 2},\underbrace{1}_{\displaystyle 1},\underbrace{2\,,2}_{\displaystyle 2},\underbrace{1\,,1}_{\displaystyle 2},...$$ Here the complexity lies in the fact that no one (yet) managed to prove that there are "equally many $1$s and $2$s" in the sense that the faction of $1$ in the first $n$ terms goes to $1/2$ as $n$ goes to $\infty$.
Edit: the comment of PM 2Ring suggest there could be nice representations. Take a look at it!
Consider a graph, and on each vertex $e$, put a certain number $n_e$ of grains of sand. Now, at each integer time, each pile that has more grains than neighbors collapse, giving one grain to each of its neighbor. Let it evolve!
Here, the darker the color, the higher the pile! I've initialized a cell with a big number a grain, and let it collapse over time. But you can choose to put two such piles and see how they interact:
If you want to consider other cellular automata, you can talk about traffic model: imagine having a circular and discrete ($n$ spots) road, on which $m$ cars are moving. They move according to the following rules: if the spot ahead is free, they move forward, otherwise they stay until next "update time". The global behavior of the cars changes dramatically when $n>2m$:
Here, as the density $p$ of occupied spots grows, we see traffic jams forming!
The possibilities are limitless!