Smooth compact complex surfaces admitting non-trivial surjective endomorphisms are classified up to isomorphism. The algebraic case was dealt with earlier by the authors. The following surfaces are listed in the non-algebraic case: a complex torus, a Kodaira surface, a Hopf surface with at least two curves, a successive blowups of an Inoue surface with curves whose centers are nodes of curves, and an Inoue surface without curves satisfying a rationality condition.
"Compact complex surfaces admitting non-trivial surjective endomorphisms." Tohoku Math. J. (2) 57 (3) 395 - 426, September 2005. https://doi.org/10.2748/tmj/1128703004