The following is motivated by a (now-deleted) MSE-question by @aglearner.

Suppose that $X\subset {\mathbb C}^n$ is an affine subvariety, equipped with the classical (Euclidean) topology. Consider the group $G= {\mathbb C}^\times$, and suppose that $G\times X\to X$ is an algebraic action. Assume, in addition, that each $G$-orbit in $X$ is closed.

**Question 1.** Is the quotient $X/G$ (in the sense of general topology) Hausdorff?

(Note that I am *not* taking the GIT quotient here.)

Hausdorffness of the quotient fails if I relax the assumption on $X$ to that of a *quasi-affine* subvariety in the following standard example: Take $X={\mathbb C}^2\setminus \{(0,0)\}$ and consider the action given by
$$
(t, (x,y)) \mapsto (tx, t^{-1}y), \quad t\in G, (x,y)\in X.
$$

The next question is a complex-analytic version of Question 1:

**Question 2.** Suppose that $X\subset {\mathbb C}^n$ is a Stein submanifold, $G$ is as above and $G\times X\to X$ is a holomorphic action with closed orbits. Is $X/G$ Hausdorff?

I expect answers to both questions to be negative (and counter-examples given by a smooth affine subvariety $X$) but cannot think of any examples.