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.
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