Could not include synthetic complex geometry - contents
The notion of complex analytic space is the notion of analytic space in complex geometry; the generalization of the notion of complex manifold to spaces with singularities.
A complex analytic test space is a common vanishing locus of a set of holomorphic functions $\mathbb{C}^n \to \mathbb{C}$. This is naturally a locally ringed space over the complex numbers $\mathbb{C}$. A complex analytic space is a locally ringed space over $\mathbb{C}$ that is locally isomorphic to such a complex analytic test space.
A smooth complex analytic space is locally isomorphic to a polydisc and hence locally contractible. See also (Berkovich, p.2).
Comparison to complex algebraic varieties (GAGA):
Introductions include
Generalization of smooth complex analytic spaces to smooth $p$-adic analytic spaces is discussed in
Discussion in higher geometry/higher algebra (derived complex analytic spaces) is in
Jacob Lurie, section 4.4. of Structured Spaces, 2008
Jacob Lurie, sections 11 and 12 of Closed Immersions