  Dual complex matrices have found applications in brain science. There are two
different definitions of the dual complex number multiplication. One is
noncommutative. Another is commutative. In this paper, we use the commutative
definition. This definition is used in the research related with brain science.
  Under this definition, eigenvalues of dual complex matrices are defined.
However, there are cases of dual complex matrices which have no eigenvalues or
have infinitely many eigenvalues. We show that an $n \times n$ dual complex
matrix is diagonalizable if and only if it has exactly $n$ eigenvalues with $n$
appreciably linearly independent eigenvectors. Hermitian dual complex matrices
are diagonalizable. We present the Jordan form of a dual complex matrix with a
diagonalizable standard part, and the Jordan form of a dual complex matrix with
a Jordan block standard part. Based on these, we give a description of the
eigenvalues of a general square dual complex matrix.
