We present a sample of 131 quasars from the Sloan Digital Sky Survey at redshifts 0.8 < z < 1.6 with double peaks in either of the high-ionization narrow emission lines [Ne V]λ3426 or [Ne III]λ3869. These sources were selected with the intention of identifying high-redshift analogs of the z < 0.8 active galactic nuclei (AGN) with double-peaked [O III]λ5007 lines, which might represent AGN outflows or dual AGN. Lines of high-ionization potential are believed to originate in the inner, highly photoionized portion of the narrow line region (NLR), and we exploit this assumption to investigate the possible kinematic origins of the double-peaked lines. For comparison, we measure the [Ne V]λ3426 and [Ne III]λ3869 double peaks in low-redshift (z < 0.8) [O III]-selected sources. We find that [Ne V]λ3426 and [Ne III]λ3869 show a correlation between line-splitting and line-width similar to that of [O III]λ5007 in other studies; and the velocity-splittings are correlated with the quasar Eddington ratio. These results suggest an outflow origin for at least a subset of the double-peaks, allowing us to study the high-ionization gas kinematics around quasars. However, we find that a non-neligible fraction of our sample show no evidence for an ionization stratification. For these sources, the outflow scenario is less compelling, leaving the dual AGN scenario as a viable possibility. Finally, we find that our sample shows an anti-correlation between the velocity-offset ratio and luminosity ratio of the components, which is a potential dynamical argument for the presence of dual AGN. Therefore, this study serves as a first attempt at extending the selection of candidate dual AGN to higher redshifts.