Let $f,g:\mathbb R\to\mathbb C$. Is there a standard notion of $f = O(g)$?
If I had to take a stab at a definition, I'd try something like
$f = O(g)$ provided there exists $M>0$ and $x_0\in\mathbb R$ such that $|f(x)| \leq M |g(x)|$ for all $x>x_0$.
where $|\cdot|$ denotes the complex mod.
You can write it like that, no problem. Even Wikipedia is on your side: it defines $f=O(g)$ as $|f|\le M|g|$ (in an appropriate domain, for some $M$).
Some people prefer writing $f=O(|g|)$, so that the function inside of $O$ is always nonnegative. But there is no mathematical difference here, only a matter of preference.