The Outflowing [OII] Nebulae of Compact Starburst Galaxies at z $\sim$ 0.5

High-velocity outflows are ubiquitous in compact, massive (M${}_{\ast }\sim$ 10${}^{11}$ M${}_{\odot }$), z $\sim$ 0.5 galaxies with extreme star formation surface densities (${\mathrm{\Sigma }}_{SFR}\sim$ 2000 M${}_{\odot }$ yr${}^{-1}$ kpc${}^{-2}$). We have previously detected and characterized these outflows using MgII absorption lines. To probe their full extent, we present Keck/KCWI integral field spectroscopy of the [OII] and MgII emission nebulae surrounding all of the 12 galaxies in this study. We find that [OII] is more effective than MgII in tracing low surface brightness, extended emission in these galaxies. The [OII] nebulae are spatially extended beyond the stars, with radial extent R${}_{90}$ between 10 and 40 kpc. The nebulae exhibit non-gravitational motions, indicating galactic outflows with maximum blueshifted velocities ranging from -335 to -1920 km s${}^{-1}$. The outflow kinematics correlate with the bursty star formation histories of these galaxies. Galaxies with the most recent bursts of star formation (within the last $<$ 3 Myr) exhibit the highest central velocity dispersions ($\sigma >$ 400 km s${}^{-1}$), while the oldest bursts have the lowest-velocity outflows. Many galaxies exhibit both high-velocity cores and more extended, slower-moving gas indicative of multiple outflow episodes. The slower, larger outflows occurred earlier and have decelerated as they propagate into the CGM and mix on timescales $>$ 50 Myr.