geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A complex Lie group is a Lie group that is a group object not just internal to smooth manifolds but in fact to complex manifolds. Hence it is a complex manifold $G$ equipped with a group structure such that both the multiplication map $G \times G \to G$ as well as the inverse map $G \to G$ are holomorphic functions.
One can also consider groups in almost complex manifolds. But every almost complex Lie group is automatically also a complex Lie group.
For $K$ a compact Lie group there is a unique connected complex Lie group $G$ such that
the Lie algebra of $G$ is the complexification of the Lie algebra of $K$:
$K$ is a maximal compact subgroup of $G$.
This $G$ is called the complexification of $K$.
(coset spaces of complex Lie groups are Oka manifolds)
Every complex Lie group and every coset space (homogeneous space) of complex Lie groups is an Oka manifolds.
See also