Given an analytic circle diffeomorphism f ∶ ℝ/ℤ → ℝ/ℤ and a complex number w, ℑw > 0, consider the quotient space of the annulus 0 \le ℑz \le ℑw, z ∈ ℂ/ℤ, by the action of f + w. This quotient space is a torus; its modulus is called the complex rotation number of f + w.

Limit values of the complex rotation number on ℝ/ℤ include a fractal set ``bubbles'': analytic loops attached to all rational points of the real axis. ``Bubbles'' is a complex analogue to Arnold tongues. The talk will include a survey of results, some ideas of proofs, and open questions.