  We consider nonsymmetric hermitian complex Hadamard matrices belonging to the
Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we
give a characterization of the eigenmatrix of a commutative nonsymmetric
association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric
hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix
is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th
roots of unity. We also give nonsymmetric association schemes X of class 6 on
Galois rings of characteristic 4, and classify hermitian complex Hadamard
matrices belonging to the Bose-Mesner algebra of X. It is shown that such a
matrix is again necessarily a Butson-type complex Hadamard matrix whose entries
are 4-th roots of unity.
