Bairstow's method loses its quadratic rate of convergence when trying to find a quadratic factor of multiplicity greater than one, although there are a couple of modifications to it which fix this.  I do not believe it loses its quadratic convergence to a single quadratic factor based on whether that quadratic factor has distinct or identical roots, as it is searching for the quadratic coefficients, and not for the roots directly.  I've never seen a reference which stated the latter specifically, but there is lots of literature on the quadratic multiplicity difficulty, and one would conjecture that if double roots also caused a problem, they would be mentioned in that.