Why do we use $z$ for representing complex number? Why do we use $z$ for representing complex number? Is there any specific reason? or just tradition
 A: This is simply convention and we could denote them any way we want but there is a motivation one might give to do this, which I'd argue is the complex plane. 
We tend to express coordinates with $x$ and $y$ (again, a convention but a quite common one) and a complex number $z$ can be written as a sum of its real part and imaginary part in the following way:
$$z=x+iy.$$
This means a complex number in a certain sense has an $x$ and a $y$ coordinate representation where you can place them in a Cartesian coordinate system with one axis being $x$ and one being $y$. I would argue it's natural notation to choose $z$ as the third letter in such a tupel but as said this all boils down to conventions being based on other conventions.
For information on this, if you don't already know, it might be worth looking at: https://en.wikipedia.org/wiki/Complex_plane
A: Actually, we do not always denote a complex number by $z=x+iy$. In the theory of Dirichlet series people prefer 
$$
s=\sigma+it,
$$
for example for the Riemann zeta function
$$
\zeta(s)=\sum_{n=1}^{\infty} n^{-s}, 
$$
for $\sigma>1$.
