So having just completed a project on complex dynamics...

One of the main sections of it dealt with theorems of analytic and holomorphic functions.

I used a lot of theorems that all started with "Let f be a holo/analytic function..." but at no point in any of the reading and research I conducted did I ever see any mention of WHY it is necessary for a function to be holomorphic.

In regards to Julia sets now,

Why must they be holomorphic? What conditions must hold in order for us to construct a Julia set? A polynomial with complex coefficients is holomorphic but why does this matter when we are dealing with Julia sets of the form $\displaystyle z^2 + c$. What theorem/definitions are we using that only applies to holomorphic functions?