Rolle's Theorem for Complex Functions We know the Rolle's theorem in valid for a real-valued function. Why can't we have Rolle's theorem in the for complex-valued functions? 
 A: If I understand, you want a theorem like 

Let $f$ be a holomorphic function on a open set $\Omega$ such that $f(a)=f(b)$ for $a,b \in \Omega$, then, there is $c$ on the segment $]a,b[$ such that $$\frac{\partial f}{\partial z}(c) =0.$$ 

This theorem is false because we can find a counterexample. Consider the holomorphic function $f: z\to e^{z}$. We have $f(0)=f(2i\pi)$ but the complex derivative, which is $e^z$, will never cancel. However we can have a kind of Rolle theorem which is the following (and it is a true statement)

Let $a$ and $b$ two distinct points of $\mathbb{C}$ and $f$ an entire function such that $f(a) =f(b)$. There are two points $c_1$ and $c_2$ on $]a,b[$ such that $$\Re(f'(c_1))=0 \quad \quad \Im(f'(c_2))=0.$$ 

It is important to note that $c_1$ and $c_2$ are not necessarily equal.
A: Here's one perspective. Below I've written a fairly well-known theorem sometimes called the Grace-Heawood theorem.

Let $P(z)$ be a complex polynomial of degree $d\geq 2$, and suppose that $P(1) = P(-1)$. Then $P$ must have a critical point $z_0$ somewhere in the closed disk of radius $\cot(\pi/d)$ centered at the origin (and this bound is sharp). 

You can interpret this as a Rolle's theorem for complex polynomials, since it guarantees a critical point "near" the points $1$ and $-1$, where $P(1) = P(-1)$. But the problem with this is we have to assume that we are dealing with polynomials of degree $d\geq 2$, and with that assumption comes the assumption that there are critical points to begin with. Also notice that as $d$ gets large $\cot(\pi/d)\to \infty$, so the critical point can actually be pretty far away from $1$ and $-1$. If you allow yourself to think of a holomorphic function as an "infinite-degree" polynomial, this would suggest the possibility of critical points being "infinitely far" from $1$ and $-1$. (Yes, I know this is more motivation than it is solid mathematics.) 
As the example of $e^z$ shows that holomorphic functions need not have critical points at all, even if they are not injective. And this destroys any chance that a Rolle's-type theorem could hold for arbitrary holomorphic functions. 
I think the best way to think about this is less something to do with complex analysis, and more something to do with working in real dimension $>1$. Topologically, simply a lot more can happen in higher dimensions. The exponential map $e^z$ discussed above is a non-trivial covering map $\mathbb{C} \to \mathbb{C}^\times$; this is made possible at least in part by the fact that there are connected, non-simply connected subsets of $\mathbb{C}$ like $\mathbb{C}^\times$ in the first place. This is not the case for $\mathbb{R}$.
