In this paper, we provide infinitely many examples of compact negatively curved almost complex, but not complex, manifolds of complex dimension 2n + 1 or 2n + 2 by using strongly pseudoconvex homogeneous domains in an almost complex manifold. Unlike the Kodaira-Thurston manifold which is the flat case, the first Betti number of the constructed manifolds is even, and their first homology group is, in fact, isomorphic to Z(4n+2) or Z(4n+4).