Intermediate value-like theorem for $\mathbb{C}$? Is there an intermediate value like theorem for $\mathbb{C}$? I know $\mathbb {C}$ isn't ordered, but if we have a function $f:\mathbb{C}\to\mathbb{C}$ that's continuous, what can we conclude about it? 
Also, if  we have a function, $g:\mathbb{C}\to\mathbb{R}$
,continuous with $g(x)>0>g(y)$ does that imply a z "between" them satisfying $g(z)=0$.
Edit:
I apologize if the question is vague and confusing. I really want to ask for which definition of between, (for example maybe the  part of the plane dividing the points), and with relaxed definition on intermediateness for the first part, can we prove any such results?

A book “First concepts of topology” by W.G. Chinn and N.E. Steenrod is devoted to intermediate value theorems for the line and the plane and their applications. The exposition is very intuitive, detailed, and illustrated. A counterpart of the intermediate value theorem for $\Bbb C$ is the main theorem of Part II [p. 86]. 

Let $f:D\to P$ be a mapping of a disk into the plane, let $C$ be the boundary circle of $D$, and let $y$ be a point of the plane not on $fC$. If the winding number of $f|C$ about $y$ is not zero, then $y\in fD$; i.e. there is a point $x\in D$ such that $fx=y$.

