We present a countable complete first order theory T which is model theoretically very well behaved: it eliminates quantifiers, is ω-stable, it has NDOP and is shallow of depth two. On the other hand, there is no countable bound on the Scott heights of its countable models, which implies that the isomorphism relation for countable models is not Borel.
Martin Koerwien. "A complicated ω-stable depth 2 theory." J. Symbolic Logic 76 (1) 47 - 65, March 2011. https://doi.org/10.2178/jsl/1294170989