Is algorithmic information theory still evolving?
A modern tweak on algorithmic information theory is algorithmic randomness which was developed intensively in the 2000s (2009-2009) and is still quite active. 

The most notorious open problem there may be whether Kolmogorov-Loveland randomness (in which martingales are computable but are allowed to bet on bits out of order) is the same as Martin-Löf randomness (in which martingales are only semicomputable, i.e., the capital function is computably approximable from below). This is known to be almost true, e.g. if $A\oplus B=\{2n:n\in A\}\cup \{2n+1:n\in B\}$ is KL-random then either $A$ or $B$ is ML-random.

An example of a recent paper in this area:

Bienvenu, Laurent, Kolmogorov-Loveland stochasticity and Kolmogorov complexity, Theory Comput. Syst. 46, No. 3, 598-617 (2010). ZBL1204.68110..