complex manifold
manifold with an atlas of charts to the open unit disk in ℂⁿ, such that the transition maps are holomorphic
Riemann surface
one-dimensional complex manifold
complex projective space
the space of lines in a complex vector space
hypercomplex manifold
smooth manifold equipped with an action of the quaternion algebra on the tangent bundle, such that there is an S² family of (integrable) complex structures