Marine heatwaves and other extreme temperature events can drive biological responses, including mass mortality. However, their effects depend on how they are experienced by biological systems (including human societies). We applied two different baselines (fixed and shifting) to a time series of North Sea water temperature to explore how slowly vs. quickly adapting systems would experience extreme temperatures. We tested if the properties of marine heatwaves and the association with atmospheric heatwaves were robust to a change in baseline. A fixed baseline produced an increase in the frequency and duration of marine heatwaves, which would be experienced as the new normal by slowly adapting systems; 7 of the 10 most severe heatwaves occurred between 1990 and 2018. The shifting baseline removed the trend in the frequency but not duration of heatwaves; the 1990s appeared as a period of change in the frequency of strong and severe heatwaves as compared to the 1980s. There were also common patterns among baselines: marine heatwaves were more frequent in late summer when temperatures peak; temperature variability was characterized by low frequency, large amplitude fluctuations (i.e., as red noise), known to drive extinction events. In addition, marine heatwaves occurred during or just after atmospheric heatwaves. Our work highlights the importance of identifying properties of marine heatwaves that are robust or contingent on a change in baseline.