Every finitely presented group is the fundamental group of a compact complex manifold of dimension $3$.
This is proven in the book [Amoros, by Amoros, Burger, Corlette, Kotschick , and Toledo : Fundamental groups of compact Kahler manifolds]manifolds, Section 4.3 Corollary 1.66 p. 19.
The rough idea of proof is the following. Let $\Gamma$ be a finitely presented group, and let $Y$ be a smooth closed oriented $4$-manifold with $\pi_1(Y) \cong \Gamma$. Then by a result of Taubes one can find a complex $3$-fold with the same fundamental group by taking the twistor space $Z$ of $X=Y \sharp n \overline{\mathbb{C} \mathbb{P}^2}$ for $n$ sufficiently large.
