  We use the ZFIRE survey (http://zfire.swinburne.edu.au) to investigate the
high mass slope of the initial mass function (IMF) for a mass-complete
(log10(M$_*$/M$_\odot$)~9.3) sample of 102 star-forming galaxies at z~2 using
their H$\alpha$ equivalent widths (H$\alpha$-EW) and rest-frame optical
colours. We compare dust-corrected H$\alpha$-EW distributions with predictions
of star-formation histories (SFH) from PEGASE.2 and Starburst99 synthetic
stellar population models. We find an excess of high H$\alpha$-EW galaxies that
are up to 0.3--0.5 dex above the model-predicted Salpeter IMF locus and the
H$\alpha$-EW distribution is much broader (10--500 \AA) than can easily be
explained by a simple monotonic SFH with a standard Salpeter-slope IMF. Though
this discrepancy is somewhat alleviated when it is assumed that there is no
relative attenuation difference between stars and nebular lines, the result is
robust against observational biases, and no single IMF (i.e. non-Salpeter
slope) can reproduce the data. We show using both spectral stacking and Monte
Carlo simulations that starbursts cannot explain the EW distribution. We
investigate other physical mechanisms including models with variations in
stellar rotation, binary star evolution, metallicity, and the IMF upper-mass
cutoff. IMF variations and/or highly rotating extreme metal poor stars
(Z~0.1Z$_\odot$) with binary interactions are the most plausible explanations
for our data. If the IMF varies, then the highest H$\alpha$-EWs would require
very shallow slopes ($\Gamma$>-1.0) with no one slope able to reproduce the
data. Thus, the IMF would have to vary stochastically. We conclude that the
stellar populations at z~2 show distinct differences from local populations and
there is no simple physical model to explain the large variation in
H$\alpha$-EWs at z~2.
