**In the following, an algebra will always mean a finite-dimensional associative commutative unital algebra (over some field $k$).**

Let $A$ be a $\mathbb{C}$-algebra. I am trying to understand how its group of units $A^{\times}$ sits topologically in $A$ relative to the locus of non-invertible elements $A\setminus A^{\times}$.

In the answer to this question at MSE, Jeremy Rickard gives a nice argument showing that every $\mathbb{R}$-algebra $A$ with connected group of units $A^{\times}$ is necessarily complex.

(Q1) Is the converse true, i.e. does every $\mathbb{C}$-algebra have connected group of units?

If not,

(Q2) Can we characterize / classify those $\mathbb{C}$-algebras with connected group of units? Or is that too much to hope for? (Note that in general the classification of finite-dimensional commutative $\mathbb{C}$-algebras is a hard problem, see for example B.Poonen's work for more details on this.)

Since $A$ *as above* is necessarily complex, it follows that $A^{\times}$ is connected abelian complex Lie group, hence $A^{\times}\simeq\mathbb{C}^n/\Gamma$ for some lattice $\Gamma\subset\mathbb{C}^n$. (In fact, by Remmert-Morimoto $A^{\times}$ is isomorphic to a product of finite number of copies of $\mathbb{C}$, of $\mathbb{C}^{\times}$, and a Cousin group.)

(Q3) Conversely, when can a connected abelian complex Lie group be realized as the group of units of some $\mathbb{C}$-algebra $A$?