Limit / Differentiability of complex functions

I am new to delta-epsilon arguments to prove the existence of limits.

Q - Find all the points on the complex plain where f(z) = |z|^2 is differentiable? i.e. I seek all z0 where lim z->z0 [f(z)-f(z0)]/[z-z0] exisits

Can someone please help me with a nice delta-epsilon argument to solve this?

I did it without using any delta-epsilon argument. Answer I got is only z0 = 0 to be the only point.

Would really appreciate is someone can help me with a rigorous argument based on delta-epsilon definition of existence of limit.