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We propose 162.2 hours (including overheads) of Q-band observations with the VLBA in order to explore the underlying nature of eight peculiar active galactic nuclei (AGN). These eight objects originate from a sample of 88 SDSS quasars identified as having broad emission lines that are velocity-offset (>1000virgulkm-s) from the host galaxy (Eracleous et al. comma 2012) comma potentially indicating a binary supermassive black hole (SMBH). However comma there are two other leading models for such shiftsdoublePoint a recoiling post-merger SMBH comma or a broad line region (BLR) gas outflow caused by a jet or accretion disk wind. We have previously followed up all 88 objects from Eracleous et al. (2012) with VLA X-band observations. We then followed up the brightest VLA-detected objects with VLBA X-band observations. Our proposed Q-band observations of the eight brightest objects will provide a critical 5-fold increase in angular resolution over the previous VLBA X-band observations comma and test the binary hypothesis down to sub-pc separations in all cases. In addition comma we request 9 orbits with the HST in order to determine the SMBH masses comma assess the host galaxies for any evidence of recent major galaxy mergers comma and measure the photometric centers that will allow more precise determination of radio-optical offsets.