an almost-complex structure

Multiplying by the imaginary unit i is a quarter-turn rotation of the complex plane: do it twice and you have rotated by 180 degrees, i.e. multiplied by -1. An almost-complex structure is what you get when you install such a quarter-turn rotation in the tangent space of every point of a real manifold, in a smoothly varying way. It is the infinitesimal shadow of being a complex manifold — it tells each tangent space how to behave like a complex vector space — without yet guaranteeing that the manifold globally looks like C^n.

Precisely, an almost-complex structure on a smooth manifold M is a smooth field of linear maps J: TM -> TM, one endomorphism J_p on each tangent space T_p M, satisfying J^2 = -identity. The condition J^2 = -1 turns each real tangent space into a complex vector space (with i acting as J), which forces the real dimension of M to be even, say 2n, and gives a canonical orientation. A manifold carrying such a J is called almost-complex. Every complex manifold has a natural J: in a holomorphic chart, J is just multiplication by i in the tangent directions. The pair (M, J) is the flexible, soft version; whether J actually comes from a genuine complex atlas is a separate, hard question.

The crucial and often-missed point is that not every almost-complex structure is integrable — most are not. An almost-complex J defines an honest complex manifold structure if and only if a certain tensor built from J, the Nijenhuis tensor, vanishes (the Newlander-Nirenberg theorem). The sphere S^6 famously admits an almost-complex structure (from the octonions) but it is non-integrable, and whether S^6 admits any integrable one is a celebrated open problem. So 'almost-complex' is strictly weaker than 'complex': it is the linear-algebra skeleton, and integrability is the missing analytic flesh.