Why the derivative of a complex function is a "Strong Demand" It is said that differentiability of a complex function is a "strong" demand why is there?
In the case of a single real variable function it is understood, as the complex number is like differentiability of two real variables function
In the case of multivariable function, is it because a complex number can be defined on different branches?
 A: If a function of two real variables is differentiable, it may not be twice-differentiable. Indeed, its derivatives might not even be continuous. 
If a function of a complex variable is differentiable, then it is infinitely differentiable, indeed analytic – it has a power series that converges to it. 
A: Apart from the analytical reason (power series expansions) explained by Gerry, there is also a purely geometric one. 
In a nutshell, real differentiability is the property that a mapping can be locally approximated by linear ones. One linear map for each point of the domain. Complex differentiability is real differentiability, combined with the strong requirement that all those linear mappings are similitudes of the complex plane, that is, combinations of a rotation and a dilation.
I recommend digging the book "Visual complex analysis" by Tristan Needham. It contains lots of pictures which explain this geometric point of view very vividly.
A: A  differentiable map $${\bf f}:\quad{\mathbb R}^2\to{\mathbb R}^2\tag{1}$$ has at each point ${\bf p}$ of its domain a Jacobian matrix $J_{\bf f}({\bf p})$.  This matrix changes from point to point, but can at any point ${\bf p}$ be any real $2\times2$-matrix. 
On the other hand, if you have a complex differentiable function $f:\>{\mathbb C}\to{\mathbb C}$ and view this function as a map of type $(1)$ then the Jacobian of this resulting ${\bf f}$ has at each point ${\bf p}$ the special form
$$J_{\bf f}({\bf p})=\left[\matrix{a&-b \cr b&a\cr}\right]\ ,$$
whereby $a+ib=f'(p)$. (This is the content of the CR equations.) In other words, for a complex differentiable function at each point we only have two real degrees of freedom in $J$, whereas for a real differentiable function we have four degrees of freedom.
