  We apply the quaternionic Jordan form to classify the hypercomplex nilpotent
almost abelian Lie algebras in all dimensions and to carry out the complete
classification of 12-dimensional hypercomplex almost abelian Lie algebras.
Moreover, we determine which 12-dimensional simply connected hypercomplex
almost abelian Lie groups admit lattices. Finally, for each integer $n>1$ we
construct infinitely many, up to diffeomorphism, $(4n+4)$-dimensional
hypercomplex almost abelian solvmanifolds which are completely solvable. These
solvmanifolds arise from a distinguished family of monic integer polynomials of
degree $n$.
