We present the discovery and high-cadence follow-up observations of SN 2018ivc, an unusual Type II supernova that exploded in NGC 1068 (D = 10.1 Mpc). The light curve of SN 2018ivc declines piecewise-linearly, changing slope frequently, with four clear slope changes in the first 40 days of evolution. This rapidly changing light curve indicates that interaction between the circumstellar material and ejecta plays a significant role in the evolution. Circumstellar interaction is further supported by a strong X-ray detection. The spectra are rapidly evolving and dominated by hydrogen, helium, and calcium emission lines. We identify a rare high-velocity emission-line feature blueshifted at ~7800 km/s (in Ha, Hb, Pb, Pg, HeI, CaII), which is visible from day 18 until at least day 78 and could be evidence of an asymmetric progenitor or explosion. From the overall similarity between SN 2018ivc and SN 1996al, the Ha equivalent width of its parent HII region, and constraints from pre-explosion archival Hubble Space Telescope images, we find that SN 2018ivc may have had a high-mass progenitor (with initial mass > 25 Msun). SN 2018ivc demonstrates the importance of the early discovery and rapid follow-up observations of nearby supernovae to study the physics and progenitors of these cosmic explosions.