  We construct 4-dimensional Riemannian Lie groups carrying left-invariant
conformal foliations with minimal leaves of codimension 2. We show that these
foliations are holomorphic with respect to an (integrable) Hermitian structure
which is not K\" ahler. We then prove that the Riemannian Lie groups
constructed are {\it not} Einstein manifolds. This answers an important open
question in the theory of complex-valued harmonic morphisms from Riemannian
4-manifolds.
