  We present measurements of the local primordial non-Gaussianity parameter
\fNLloc from the clustering of 343,708 quasars with redshifts 0.8 < z < 2.2
distributed over 4808 square degrees from the final data release (DR16) of the
extended Baryon acoustic Oscillation Spectroscopic Survey (eBOSS), the largest
volume spectroscopic survey up to date. Our analysis is performed in Fourier
space, using the power spectrum monopole at very large scales to constrain the
scale dependent halo bias. We carefully assess the impact of systematics on our
measurement and test multiple contamination removal methods. We demonstrate the
robustness of our analysis pipeline with EZ-mock catalogues that simulate the
eBOSS DR16 target selection. We find $f_\mathrm{NL}=-12\pm 21$ (68\%
confidence) for the main clustering sample including quasars with redshifts
between 0.8 and 2.2, after exploiting a novel neural network scheme for
cleaning the DR16 sample and in particular after applying redshift weighting
techniques, designed for non-Gaussianity measurement from large scales
structure, to optimize our analysis, which improve our results by 37\%.
