  We develop a complex scaling method for describing the resonances of deformed
nuclei and present a theoretical formalism for the bound and resonant states on
the same footing. With $^{31}$Ne as an illustrated example, we have
demonstrated the utility and applicability of the extended method and have
calculated the energies and widths of low-lying neutron resonances in
$^{31}$Ne. The bound and resonant levels in the deformed potential are in full
agreement with those from the multichannel scattering approach. The width of
the two lowest-lying resonant states shows a novel evolution with deformation
and supports an explanation of the deformed halo for $^{31}$Ne.
