Given an analytic circle diffeomorphism $f \colon \mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$ and a complex number $w$, $\operatorname{Im} w>0$, one can construct a complex torus by gluing the borders of the annulus $z \in \mathbb{C}/\mathbb{Z}, 0< \operatorname{Im} z < \operatorname{Im} w,$ via $f+w$. This is a construction due to V.Arnold (1978).

The modulus of this torus is called the complex rotation number of $f+w$. It depends holomorphically on $w \in \mathbb{H}$. As $w \to \mathbb{R}$, the annulus degenerates. The limit behaviour of the complex rotation number as $w \to \mathbb{R}$ is related to the (usual) rotation number of $f+w$, $w \in \mathbb{R}$, and for rational rotation numbers, is described by a fractal-like set (bubbles).

I shall give a survey on the topic, based on the results of V.Moldavskij, Yu.Ilyashenko, J.Lacroix, X.Buff and myself.