  We analyze a sample of 25 [Ne V] $\lambda$3426 emission-line galaxies at
$1.4<z<2.3$ using Hubble Space Telescope/Wide Field Camera 3 G102 and G141
grism observations from the CANDELS Lyman-$\alpha$ Emission at Reionization
(CLEAR) survey. [Ne V] emission probes extremely energetic photoionization
(97.11-126.21 eV), and is often attributed to energetic radiation from active
galactic nuclei (AGN), shocks from supernova, or an otherwise very hard
ionizing spectrum from the stellar continuum. In this work, we use [Ne V] in
conjunction with other rest-frame UV/optical emission lines ([O II]
$\lambda\lambda$3726,3729, [Ne III] $\lambda$3869, H$\beta$, [O III]
$\lambda\lambda$4959,5007, H$\alpha$+[N II] $\lambda\lambda$6548,6583, [S II]
$\lambda\lambda$6716,6731), deep (2--7 Ms) X-ray observations (from Chandra),
and mid-infrared imaging (from Spitzer) to study the origin of this emission
and to place constraints on the nature of the ionizing engine. The majority of
the [Ne V]-detected galaxies have properties consistent with ionization from
AGN. However, for our [Ne V]-selected sample, the X-ray luminosities are
consistent with local ($z\lesssim 0.1$) X-ray-selected Seyferts, but the [Ne V]
luminosities are more consistent with those from $z\sim 1$ X-ray-selected QSOs.
The excess [Ne V] emission requires either reduced hard X-rays, or a $\sim$0.1
keV excess. We discuss possible origins of the apparent [Ne V] excess, which
could be related to the ``soft (X-ray) excess'' observed in some QSOs and
Seyferts, and/or be a consequence of a complex/anisotropic geometry for the
narrow line region, combined with absorption from a warm, relativistic wind
ejected from the accretion disk. We also consider implications for future
studies of extreme high-ionization systems in the epoch of reionization ($z
\gtrsim 6$) with JWST.
