For z = r e^{ if} the squared value is z^{ 2} = r^{ 2} e^{ 2if}. Therefore the map f_{c} wraps twice the complex plane z onto itself (with quadratic deformation of r and displacement by c).
This is the simplest Julia set for c = 0 + 0i . As since
for  z_{o}  < 1, z_{n} converges to the fixed point z = 0, for  z_{o}  > 1, z_{n} go to infinity and for  z_{o}  = 1, z_{n} rotates and stays on the same circle  z  = 1 . The circle is the Julia set J(0) . It is evident, that the circle is invariant under f_{o} = z^{ 2}. 
In the applet below you can trace quadratic map dynamics.
The white square is mapped in the region with inverted colors.
You see thet Julia set is similar in both regions ("inverted" square
is deformed due to ~ε^{ 2} and higher terms in the
Taylor's formula).
Controls: Drag the white square by mouse to move it.
Z are coordinates the white square center,
R is its size, dz is the scale of the visible region.
As ususal press <Enter> to set new parameters values. E.g. set
Cr=Ci=0 to test J(0) dynamics.
You can see below selfsimilarity of "dendrite" and "midgets" Julia sets.

It is not difficult to imagine how f_{1} maps points of the
J(1) set from the Re(z) > 0 (or Im(z) > 0) halfplane
onto the whole J(1).
Squaring "moves" J(1) to the right (the lower picture) and after addition of c = 1 the Julia set returns into its original position. Note, that the two points a are mapped into one point a'. Moreover for c = 1.77289 the biggest J(1) midget located at z = 0 (to the right below) in a similar way is mapped (twice) into a small one (in the white square to the left). Renormalization theory and the Julia midgets scaling will be discussed later. 
It is easy to see, that the "cauliflower" J(0.35) set has the same
"squaring" symmetry.
Two pictures below illustrate the squaring transformation for the Douady rabbit. The line RR cuts the creature in two symmetric halfs and each part is mapped by f_{c} into the whole Jset. Two points R pass into R' one. A "unit" square is used instead of a circle.  