Does it ever make sense to have the window open with AC on?

If it's cooler outside than in, then in general opening the windows and having the AC on will cool things off faster than just one or the other.

Caveats might include the AC output being near a window and it being cooler inside near the window, or humidity. But absent that, sure, both will cool it down faster.

Once the inside temperature drops below outside, though, probably want to close the windows or turn off the AC.

I'd also add that if your AC has a ventilation-only mode that can pull outside air in --- a window unit or ducted unit probably does, split mini won't, portable may not --- that'll help cool the inside and be more energy-efficient than running the compressor, if you don't care about getting inside colder than ambient temperatures. Might also consider putting a box fan in a window or two, which could also drastically increase air turnover rate.

I live in a pretty comfortable climate and normally always have an open window and a small fan near the ceiling blowing air out the window, to have more airflow than would normally be the case with purely-passive ventilation.

If you're in an environment that doesn't get much humidity --- the Southwest in the US is a good example --- you might also consider an evaporative cooler, which won't give you the potential very cold temperatures of an air conditioner (given enough power), but will cool things below ambient temperatures without needing much power.

You provided an image of a mini split. They are so efficient, if it is colder outside than in it is pretty negligible that it will be any less efficient. Unless the window is near the outside unit and the heat just comes back in. But you need to take into account the humidity, if it's much more humid outside you definitely should keep the windows shut.