The Spilhaus projection displays the world ocean as a continuous water mass around Antarctica with the other (highly distorted) continents acting as the map's boundary. I think it would be a great addition to cartopy's functionality especially for oceanographers.

Here is a nice write up of the history of the projection with potential starting points for the implementation: ArcGIS storymap on Spilhaus projection

(image from aforementioned ArcGIS link)

If I see it correctly, we would need the Adams World Projection in a Square II implemented in PROJ (ref. Adams, O. S. (1929). Conformal Projection of the Sphere Within a Square. Washington: [U.S. Coast and Geodetic Survey Special Publication 153: ftp://ftp.library.noaa.gov/docs.lib/htdocs/rescue/cgs_specpubs/QB275U35no1531929.pdf). Is that correct? Further, can cartopy rotate the resulting projection 45 degrees or does that need to be implemented on the PROJ side?