We consider the zero-temperature coarsening in the Ising model in two dimensions where the spins interact within the Moore neighborhood. The Hamiltonian is given by H=-i,jSiSj-κi,j′SiSj′, where the two terms are for the first neighbors and second neighbors, respectively, and κ≥0. The freezing phenomenon, already noted in two dimensions for κ=0, is seen to be present for any κ. However, the frozen states show more complicated structure as κ is increased; e.g., local antiferromagnetic motifs can exist for κ>2. Finite-sized systems also show the existence of an isoenergetic active phase for κ>2, which vanishes in the thermodynamic limit. The persistence probability shows universal behavior for κ>0; however, it is clearly different from the κ=0 results when a nonhomogeneous initial condition is considered. Exit probability shows universal behavior for all κ≥0. The results are compared with other models in two dimensions having interactions beyond the first neighbor. © 2017 American Physical Society.