**Question: **
Is $f : C → C$ given by $f(z) = z^2 + z|z|^2$ differentiable at $z = 0$? If so, what is $f'(0)$?
Does $f^{n}(0)$ exist for $n ≥ 2$?
The first two questions are quite clear. It is differentiable at $z=0$ and $f'(0) = 0$
The problem is, I can't figure out how to approach the third part. The first approach of figuring out whether $f^{n}(0)$ exist for $n ≥ 2$ I thought was to figure out if $f$ is infinitely differentiable at points including $(0,0)$.
So I need to set an equation $lim_{h \rightarrow 0} (f'(0+h) - f'(0))/h$.
However, as it is clear, $f(z)$ is not differentiable at point other than $(0,0)$($\because$ $|z|^2$ is only differentiable at $(0,0)$) so $f'(0+h)$ can not be defined, which makes that limit equation invalid.
Does that mean the answer for the third question is "$f^{n}(0)$ does not exists for $n≥2$"? Or is there some part that wrong on my approach such that $f^{n}(0)$ exists.
Your conclusion is indeed the right one.
In order to speak of $f^{\prime\prime}(0)$, $f^\prime$ has to be defined in a neighborhood of $0$. As $z \mapsto z^2$ is holomorphic, $g^\prime(z)$ has to be defined in a neighborhood of $0$ where $g(z) = z \vert z \vert^2$.
But that can't be as if it was the case, $\frac{g(z)}{z}$ would be holomorphic for $z \neq 0$ which is not the case (it is a map taking only real values).
Conclusion: $f^{\prime}$ is not defined in a neighborhood of zero and $f^{\prime\prime}(0)$ can't be defined. As well as obviously higher derivatives.