Overlapping images in escape time fractals

[Andrew Jenner]

I've been playing about with drawing escape time fractals using
different formulae. I've noticed that some formulae give images which
seem to have a strange feature - in some places it looks like there
are two overlapping images, with one image showing through in areas
of the other.

For example, here is an image I made using the formula
z <- (z+1)*(z+c)*(z+i)
http://www.reenigne.org/misc/z1zczi.png (2.5Mb image)

Can anybody help me to understand what is going here? Are there really
two iterative process going on hidden in one formula or is it an
illusion? Does this phenomenon have a name that I could search for to
find out more about it?

Many thanks.