Is the complex structure of $\mathbb CP^n$ unique?

Let me write this too long comment as an answer.

As abx says, what we do know is

Theorem 1. If a Kähler manifold $X$ is homeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to it.

This is due to Hirzebruch and Kodaira for $n$ odd (but with the strongest assumption for $X$ to be diffeomorphic to $\mathbb{CP}^n$, then relaxed to homeomorphic after work of Novikov), and to Yau for $n$ even.

For $n=2$, a stronger result holds, still proved by Yau, namely

Theorem 2. If a compact complex surface $S$ is homotopy equivalent to $\mathbb{CP}^2$, then it is biholomorphic to it.

In dimension $n\le 6$, we have instead a result due to Libgober-Wood which is stronger than Theorem 1, but weaker than Theorem 2, that states

Theorem 3. A compact Kähler manifold of complex dimension $n\le 6$ which is homotopy equivalent to $\mathbb{CP}^n$ must be biholomorphic to it.

You can find all this (and much more) on the very beautiful survey by V. Tosatti available here.

I would also like to mention an interesting related result of T. Fujita (which is not cited in the referenced survey article). "On topological characterizations of complex projective spaces and affine linear spaces", Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 5, 231–234.

Theorem: Let X be a smooth Fano n-fold with cohomology ring isomorphic to $H^{*}(\mathbb{CP}^n,\mathbb{Z})$ and $n \leq 5$. Then $X \cong \mathbb{CP}^n$.