  A new set of $ h(1) \oplus su(2)$ vector algebra eigenstates on the matrix
domain is obtained by defining them as eigenstates of a generalized
annihilation operator formed from a linear combination of the generators of
this algebra which eigenvalues are distributed as the elements of a square
complex normal matrix. A combined method is used to compute these eigenstates,
namely, the method of exponential operators and that of a system of first-order
linear differential equations. We compute these states for all possible
combination of generators and classify them in different categories according
to a generalized commutation relation as well as according to the value of a
characteristic parameter related to the $su(2)$ algebra eigenvalues. Proceeding
in this way, we found a subset of generalized vector coherent states in the
matrix domain which can be easily separated from the general set of
Schr\"odinger-Robertson minimum uncertainty intelligent states. In particular,
for a special choice of the matrix eigenvalue parameters we found the so-called
vector coherent states with matrices associated to the Heisenberg-Weyl group as
well as a generalized version of them, and also a direct connection with the
coherent state quantization of quaternions.
