  We study the rational Bianchi newforms (weight 2, trivial character, with
rational Hecke eigenvalues) in the LMFDB that are not associated to elliptic
curves, but instead to abelian surfaces with quaternionic multiplication. Two
of these examples exhibit a rather special kind of behaviour: we show they
arise from twisted base change of a classical newform with nebentypus character
of order 4 and eight inner twists.
