Is it always possible to split a complex function into the sum of its real + imaginary parts? I'm looking for any kind of proof or at least an intuitive explanation. I am curious because I want to know:

if I have an unknown complex function that I want to manipulate, can I assume it has the structure a(x) + ib(x).

It seems to me this should definitely be possible. I'm hoping someone can provide a lucid argument that makes the answer seem stupidly obvious
Of course you can.  The real and imaginary parts of a complex number uniquely determine it.  You can identify $\mathbb{C}$ with the plane $\mathbb{R}^2$ via $a + ib \mapsto (a,b)$.  

Let $X$ be a set, and let $f$ be a function from $X$ to $\mathbb{C} = \mathbb{R}^2$.  Let $\pi_1: \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function $\pi(a,b) = a$, and let $\pi_2: \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function $\pi_2(a,b) = b$.

Let $u: X \rightarrow \mathbb{R}$ be the composition $\pi_1 \circ f$, and let $v: X \rightarrow \mathbb{R}$ be the composition $\pi_2 \circ f$.  Then obviously 

$$f(x) = (u(x),v(x))$$

for any $x \in X$.  Or in the usual notation of complex numbers,

$$f(x) = u(x) + i v(x)$$

for all $x \in X$.  This is written more succinctly as $f = u + iv$.
Hint: Any complex number $z$ can be written in the form $re^{i\theta}$...