Complex analytic quantum groups

We discuss a missing link in quantum group theory — quantum analogues of complex Lie groups. As such analogues, I propose to take topological Hopf algebras with a finiteness condition (holomorphically finitely generated or HFG for short). This approach is not directly related to $C^*$-algebraic quantum groups (at least for now) but is an alternative view. Nevertheless, the topic seems to offer a wide range of research opportunities.
Our focus is on examples, such as analytic forms of some classical quantum groups (a deformation of a solvable Lie group and Drinfeld-Jimbo algebras). I also present some general results: 1) the category of Stein groups is anti-equivalent to the category of commutative Hopf HFG algebras; 2) If $G$ is a compactly generated Lie group, then the cocommutative topological Hopf algebra $\widehat{A(G)}$ is HFG. When, in addition, $G$ is connected linear, the structure of $\widehat{A(G)}$ can be described explicitly.
I also plan to discuss briefly holomorphic duality in the sense of Akbarov (which is parallel to Pontryagin duality).