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

Their Corollary 6.2 says that if you expand an algebraically closed field K by a predicate for a multiplicative subgroup G with the Mann property (equivalently, the Mordell-Lang property, i.e. the induced structure on the subgroup is only the group structure), then the expanded structure has the same kind of stability as the group structure on the subgroup.

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