Are complex differentiable function in a point analytic?

Complex differentiability at a point $z_0$ doesn't imply anything about complex differentiability at any other point. A function like $z\mapsto z\cdot \overline{z}$ is complex differentiable only at $z_0 = 0$, but it is infinitely often real differentiable, even real-analytic, on the whole plane. An example like $g(z) = \overline{z}(1-\lvert z\rvert^2)$ shows that a (real-analytic) function can be complex differentiable at each of a set of non-isolated points without being (complex) analytic anywhere.

There are some criteria that ensure a function is analytic at a point $z_0$ without a priori demanding that it be complex differentiable on a neighbourhood of $z_0$, for example it is an easy consequence of Morera's theorem that a continuous function on the unit disk $D$ is analytic at $0$ if it is complex differentiable on $D\setminus \mathbb{R}$, but such criteria require far far stronger hypotheses than just complex differentiability at a "small" set of points.