Complex powers [of numbers]

The pictures below show successive powers of complex numbers z with digits extracted according to

(2 d[Re[#], w] + d[Im[#], w]) & [z^t]

d[x_, w_] := If[x < 0, 1 - d[-x, w], IntegerDigits[x, 2, w]]

Non-trivial cases of complex number multiplication never correspond to local cellular automaton operations. (Compare page 933.)