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We report molecular gas observations of IRAS 20100-4156 and IRAS 03158+4227, two local ultraluminous infrared galaxies (ULIRGs) hosting some of the fastest and most massive molecular outflows known. Using ALMA and PdBI observations, we spatially resolve the CO(1-0) emission from the outflowing molecular gas in both and find maximum outflow velocities of $$ v_{\rm max} \sim 1600$$ and $$\sim 1700$$ km/s for IRAS 20100-4156 and IRAS 03158+4227, respectively. We find total gas mass outflow rates of $$\dot M_{\rm OF} \sim 670$$ and $$\sim 350$$ Msun/yr, respectively, corresponding to molecular gas depletion timescales $$\tau^{\rm dep}_{\rm OF} \sim 11$$ and $$\sim 16$$ Myr. This is nearly 3 times shorter than the depletion timescales implied by star formation, $$\tau^{\rm dep}_{\rm SFR} \sim 33$$ and $$\sim 46$$ Myr, respectively. To determine the outflow driving mechanism, we compare the starburst ($$L_{*}$$) and AGN ($$L_{\rm AGN}$$) luminosities to the outflowing energy and momentum fluxes, using mid-infrared spectral decomposition to discern $$L_{\rm AGN}$$. Comparison to other molecular outflows in ULIRGs reveals that outflow properties correlate similarly with $$L_{*}$$ and $$L_{\rm IR}$$ as with $$L_{\rm AGN}$$, indicating that AGN luminosity alone may not be a good tracer of feedback strength and that a combination of AGN and starburst activity may be driving the most powerful molecular outflows. We also detect the OH 1.667 GHz maser line from both sources and demonstrate its utility in detecting molecular outflows.