We investigate conditions under which a statistical manifold M (with a Riemannian metric g and a pair of torsion-free conjugate connections ∇, ∇ ∗ ) can be enhanced to a (para-)Kähler structure. Assuming there exists an almost (para-)complex structure L compatible with g on a statistical manifold M (of even dimension), then we show (M, g, L, ∇ ) is (para-)Kähler if ∇ and L are Codazzi coupled. Other equivalent characterizations involve a symplectic form ω≡ g(L·, · ). In terms of the compatible triple (g, ω, L), we show that (i) each object in the triple induces a conjugate transformation on ∇ and becomes an element of an (Abelian) Klein group; (ii) the compatibility of any two objects in the triple with ∇ leads to the compatible quadruple (g, ω, L, ∇ ) in which any pair of objects are mutually compatible. This is what we call Codazzi-(para-)Kähler manifold  which admits the family of torsion-free α -connections (convex mixture of ∇, ∇ ∗ ) compatible with (g, ω, L). Finally, we discuss the properties of divergence functions on M× M that lead to Kähler (when L= J, J2= - id ) and para-Kähler (when L= K, K2= id ) structures.