QGIS » QGIS Application

If you undock the "inputs" pane (etc) it will stay open when you close the modeller.
After that, things can get weird, although I haven't tested thoroughly enough to detail exactly what happens in what situations (perhaps it is partly operating system dependent - I tested briefly on Windows and with a slightly old nightly on Ubuntu under WSL):
- you can't close the floating panes (need to redock them and then restart qgis if there was more than one copy of one - see below).
- if you reopen the modeller you might be able to use the floating pane that is already open (or more likely I was just tricked by a new copy of the floating pane opening above an old one), but
- you might get another instance of the floating pane, in which case you can't use the first instance. The first instance will also stay open even if you dock the second one.