Do Cauchy Riemann conditions lead to saddle points everywhere in the domain? I'm starting to learn about complex differentiability. I understand that given a complex function $f = u(x,y)+iv(x,y)$, the Cauchy Riemann equations lead to the condition that $\partial^2u/\partial x^2 = -\partial^2u/\partial y^2$ at every point where the function is differentiable. 
Now, this is the condition that is also satisfied by a saddle $x^2 - y^2$, or a generic saddle like figure: $x^{2n} - y^{2n}$. 
Given a complex function that is differentiable everywhere over a continuous domain, obviously the condition should also be satisfied for every point in the domain. But the condition means that the point (for which the condition is met) is actually a saddle point. What I don't get is: how can every point in a domain be a saddle point? What would such a surface look like? I don't understand how every point and all its infinitestimally adjacent points (in said domain) all be saddles - if a point is a saddle, its adjacent points cannot be saddles as well. So, how does this fit in with the function being differentiable over the entire domain?
I'm sure I'm missing something obvious, but I seem to have confused myself nicely. Any pointers would be much appreciated. Cheers! 

It depends a bit upon what you mean by a saddle point?
A critial point for $u$  is a point for which $\partial_x u=\partial_y u = 0$.
By C-R this also means that  $\partial_x v=\partial_y v = 0$. So real and imaginary part have simultaneous critical points. 
Normally (at least what I am used to) one only associates a saddle point with a critical point. But in that case indeed it has to be a saddle point. A way to see this in terms of differential geometry is to note that the Hessian $H_u(x,y)$ (the 2nd derivative of $u$) has
zero trace $${\rm tr} (H) = \frac{\partial^2 u}{\partial x^2} + 
\frac{\partial^2 u}{\partial y^2}=0, $$ so that the two (real) eigenvalues have opposite signs. Another interpretation is in terms of the max principle for harmonic functions: No harmonic function can have a local max unless being constant (on a connected domain).
