Does $\,x>0\,$ hint that $\,x\in\mathbb R\,$? 
Does $x>0$ suggest that $x\in\mathbb R$?

For numbers not in $\,\mathbb R\,$ (e.g. in $\mathbb C\setminus \mathbb R$), their sizes can't be compared. 
So can I omit "$\,x\in\mathbb R\,$" and just write $\,x>0\,$?
Thank you.

It really depends on context. But be safe; just say $x > 0, x\in \mathbb R$. 
Omitting the clarification can lead to misunderstanding it. Including the clarification takes up less than a centimeter of space. Benefits of clarifying the domain greatly outweigh the consequences of omitting the clarification. 
Besides one might want to know about rationals greater than $0$, or integers greater than $0$, and we would like to use $x \gt 0$ in those contexts, as well. 
ADDED NOTE: That doesn't mean that after having clarified the context, and/or defined the domain, you should still use the qualification "$x\in \mathbb R$" every time you subsequently write $x \gt 0$, in a proof, for example. But if there's any question in your mind about whether or not to include it, error on the side of inclusion.
