Technical data for this type of fractal : click here (C01-01).

Here :
fn1() = sqr(), fn2() = ident(), fn3 = ident(), fn4 = acosh()
real(p1) = 0.5, imag(p1) = 128, real(p2) = -3, imag(p2) = 0, real(p3) = 0, imag(p3) = 0

If real(p1) = 0.75 instead of 0.5 and real(p2) = 0 instead of -3, the shape of the fractal looks like this (with another palette of colors) :



If imag(p1) = 256 instead of 128 and real(p2) = -3 instead of 0,
the curious shape of the fractal looks like this (with the same palette of colors and a zoom on the center of the above fractal) :