Proving a complex function is surjective Is this a valid proof of surjectivity?
$g: \mathbb{C} \rightarrow \mathbb{C}
\\
g(z) = z^2 + z
$
let $
z = a + bi 
\\
w = z^2 + z
\\
w = (a + bi)^2+(a+bi)
\\
Rearranging:
\\
w - a^2 + b^2 - 2abi = a + bi = z
$
It follows:
$g(w - a^2 + b^2 - 2abi) = z$
Therefore surjective.

No, it is not correct. Asserting that $g$ is surjective means that, for each $w\in\Bbb C$, there is some $z\in\Bbb C$ such that $g(z)=w$. You found no such $z$.
Note that $g(z)=w\iff z^2+z=w$ and that this is a quadratic equation. You can then apply the quadratic formula. To be more precise: let $s$ be a square root of $1+4w$, and then $z=\frac{-1\pm s}2$ is such that $g(z)=w$.
