geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
analytic geometry (complex, rigid, global)
geometry+analysis/analytic number theory
analytic space, analytic variety, Berkovich space
In complex geometry one studies complex manifolds as a special case of how in general differential geometry one studies more generally smooth manifolds.
In complex analytic geometry one studies, more generally, complex analytic spaces.
complex analytic geometry is closely related to algebraic geometry over the complex numbers. See at GAGA for more on this.
complex manifold, almost complex manifold
higher complex analytic geometry, derived complex analytic space
Daniel Huybrechts Complex geometry - an introduction. Springer (2004). Universitext. 309 pages. (pdf)
Claire Voisin, Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3