Hi Dima,
The expansion of the complex field by a predicate for the set of integer powers of 2 is an example. This follows from the results of Günaydin and Van den Dries in
"The Fields of Real and Complex Numbers with a Small Multiplicative Group" http://dx.doi.org/10.1017/S0024611506015747
In particular, see their Corollary 6.2. In the example, the structure induced on 2^Z is just the (multiplicative) group structure (by "Mordell-Lang for G_m"), and as a group 2^Z is isomorphic to the additive group of the integers, which is superstable, non-omega-stable.
Also, since coloured fields were mentioned, there's a version of Poizat's green field where the coloured group is elementarily equivalent to the additive group of the integers and the structure is then superstable, non-omega-stable. This is in my (J.D. Caycedo) thesis, Section 6.5, you can find it here: http://home.mathematik.uni-freiburg.de/caycedo/thesis
JD