Abstract: An almost complex manifold is a manifold M with a complex structure J on the fibers of the tangent bundle TM. A C^infty mapping is called J-holomorphic if its differential is complex linear at each point of the source. For technical reasons, our manifolds and maps are real analytic. A "holomorphic shadow" is the image of a J-holomorphic mapping from a compact complex manifold. We explore the geometry of almost complex manifolds by means of model theory. In model theory, a "structure" is an infinite set D together with a collection of subsets of D^n closed under intersections, complements, projections and their inverses, and containing the diagonals. A complex manifold and its subvarieties give rise to a structure. This structure satisfies some "dimension axioms". B. Zilber called a structure that satisfies these axioms a "Z-structure". We expect that an almost complex manifold with its holomorphic shadows and diagonals gives a Z-structure.