Opal Spiral

Newton's method applied to f(z) = z^(2 + i) - z on the principal branch. The root condition z^(1 + i) = 1 puts a whole family on the spiral r = exp(theta), but the principal branch keeps only z = 1 and z = -e^pi, with the origin as a third attractor. The spiral survives anyway, since |z^(1 + i)| = exp(ln r - theta) is constant along exactly that curve. The straight white horizon is the branch cut, where |f| jumps by e^(2 pi), and the cusped arcs are its preimages.