Does it make sense to talk about $ O(z)$ if $z$ complex? Does it make sense to talk about $ O(z)$ if $z$ complex? I would have thought that the usual definition wouldn't hold, since doesn't the fact that we don't have an order on $\mathbb{C}$ change things? Do we have to talk about $O(|z|)$ instead? 
For instance, could we say that
$$
Az^2 + Bz = O(|z|)
$$
for $A, B, z \in \mathbb{C}$, in the same way that we would say that it is $O(z)$ if we were just working in $\mathbb{R}$?
 A: 
Does it make sense to talk about $O(z)$ if $z$ complex?

It depends on how you define $O$. Here are the two slightly different definitions:

*

*$f(x) = O\bigl( g(x)\bigr)\iff \exists\ C > 0\ \exists\ x_0 > 0\ \forall\ x > x_0: |f(x)| \le C\cdot g(x)$

*$f(x) = O\bigl( g(x)\bigr)\iff \exists\ C > 0\ \exists\ x_0 > 0\ \forall\ x > x_0: |f(x)| \le C\cdot|g(x)|.$
Note that the difference here is about $g$ is embedded in absolute value signs or not. If we let $g$ be a complex valued function, the first definition doesn't make sense for non-positive functions, since the complex numbers are not ordered, whereas the latter definition works for all functions.
History
The symbol $O$ was first introduced by number theorist Paul Bachmann in 1894, in the second volume of his book Analytische Zahlentheorie. The number theorist Edmund Landau adopted it and defined it more rigorously. So the historical definition is 1. since at the time there was no need to define $O$ for negative or complex functions.
Today
Now there is no leading definition. I looked at a few Wikipedia articles: English, German, French, Spanish, Italian, Portuguese and Russian.
All articles, except the English one, use the second definition.
Solution
If you are writing a paper, you may define the Big-$O$ notation you are using in the preliminaries. The other possibility is to always write the absolute value, which makes both definitions equivalent.
