We investigate the notion of complex rotation number which was introduced by V.I.Arnold in 1978. Let $f: mathbb R/mathbb Z o mathbb R/mathbb Z$ be an orientation preserving circle diffeomorphism and let $omega in mathbb C/mathbb Z$ be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus ${z in mathbb C/mathbb Z mid 0< Im(z)< Im({omega})}$ via the map $f+{omega}$. This complex torus is isomorphic to $mathbb C/(mathbb Z+{ au} mathbb Z)$ for some appropriate ${ au} in mathbb C/mathbb Z$. According to Moldavskis (2001), if the ordinary rotation number $operatorname{rot} (f+omega_0)$ is Diophantine and if ${omega}$ tends to $omega_0$ non tangentially to the real axis, then ${ au}$ tends to $operatorname{rot} (f+omega_0)$. We show that the Diophantine and non tangential assumptions are unnecessary: if $operatorname{rot} (f+omega_0)$ is irrational then ${ au}$ tends to $operatorname{rot} (f+omega_0)$ as ${omega}$ tends to $omega_0$. This, together with results of N.Goncharuk (2012), motivates us to introduce a new fractal set, given by the limit values of ${ au}$ as ${omega}$ tends to the real axis. For the rational values of $operatorname{rot} (f+omega_0)$, these limits do not necessarily coincide with $operatorname{rot} (f+omega_0)$ and form a countable number of analytic loops in the upper half-plane.
