Does there exist a complex function which is differentiable at one point and nowhere else continuous?

Let $g \colon \mathbb{C} \rightarrow \mathbb{C}$ be a function that is discontinuous everywhere and bounded. Consider $f(z) := z^2 g(z)$. Then $f$ is continuous only at $z = 0$, and $f$ is differentiable at $z = 0$ as

$$ \lim_{z \to 0} \frac{f(z) - f(0)}{z} = \lim_{z \to 0} z g(z) = 0.$$

Consider $f(z)=|z|^2\cdot \mathbf 1_{\Bbb Q[i]}(z)$