Let $f(z):=2|z|^2-\overline{z}^2$ where $z\in \mathbb{C}$
At what points is f complex differentiable?
I can assume $\lim_{h\to0} \frac{\overline{h}}{h}$ does not exist.
I have $$\lim_{h\to0} [2z\frac{\overline{h}}{h} + 2\overline{z}+2\overline{h}-2\overline{z}\frac{\overline{h}}{h}-\frac{\overline{h}^2}{h}]$$
So I get f is complex differentiable at 0.
But when I use Cauchy - Riemann equations.
$F(x,y)=(x^2+3y^2,2xy)$ $$DF(x,y)= \begin{vmatrix} 2x&6y\\ 2y&2x\\ \end{vmatrix} $$ I get y=0 so is differentiable at all real numbers? Which one is right?
(b) Is f holomorphic anywhere? Do I use the CR equations to answer this?
In your first computation, you want to know when that limit exists as $h\to 0$. It certainly exists when $z=0$, but, in fact, it exists whenever $z-\bar z = 0$, i.e., on the real axis.
So far as your second is concerned, what is your textbook's definition of holomorphic? Does it require complex differentiability on a neighborhood?