  In many situations, in all branches of physics, one encounters power-like
behavior of some variables which are best described by a Tsallis distribution
characterized by a nonextensivity parameter $q$ and scale parameter $T$.
However, there exist experimental results which can be described only by a
Tsallis distributions which are additionally decorated by some log-periodic
oscillating factor. We argue that such a factor can originate from allowing for
a complex nonextensivity parameter $q$. The possible information conveyed by
such an approach (like the occurrence of complex heat capacity, the notion of
complex probability or complex multiplicative noise) will also be discussed.
