Non-Hausdorff orbit space of a covering action

On ${\mathbb{R}}^2\setminus \{0\}$, the powers of $T(x,y)=(2x,y/2)$ act freely with locally disjoint translates. Nevertheless $(2^{-n},1)\to (0,1)$ while $T^n(2^{-n},1)\to (1,0)$, so the two distinct limiting orbits cannot be separated in the quotient.

Lifting the hyperbolic-scaling action to the universal cover of ${\mathbb{C}}^{\ast }$ preserves the two inseparable limiting orbits. The covering surface is biholomorphic to $\mathbb{C}$, so a simply connected Riemann surface can still have a non-Hausdorff quotient by a covering space action of homeomorphisms.