So what's so special about f_c(z) = z^2+c?

Well, there's sort of two things:
  1) It's a family of functions (like f_λ(x) = λ x (1-x) before)
  2) The dynamics of any quadratic is represented by some f_c.

Julia[3z (1 - z), z, Resolution → 400, Axes → True] ;

[Graphics:../HTMLFiles/index_37.gif]

Julia[z^2 - 3/4, z, Axes → True] ;

[Graphics:../HTMLFiles/index_39.gif]

In general, the Julia set of g(z)=α  z^2+β  z+γ will look exactly like the Julia set of  f_c(z) = z^2+ c, where c=(2 β-β^2+4 α γ)/4!

α = Random[Real, {-1, 1}] ;

β = Random[] + Random[]  ;   

γ = Random[] ;

g = α z^2 + β z + γ

0.713644 + (0.609544 + 0.450464 ) z - 0.930875 z^2

Julia[g, z, Axes → True] ;

[Graphics:../HTMLFiles/index_50.gif]

c = (2β - β^2 + 4α γ)/4

-0.401698 + 0.0879433 

Julia[z^2 + c, z, Axes → True] ;

[Graphics:../HTMLFiles/index_54.gif]

Thus we're essentially studying the behavior of all possible quadratic functions.

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