Hi Dscat,
Thank you for the question.
No, it is impossible not to hide completed notes as it’s by design and deeply hard-coded (my strong preference tbh).
The app assumes that you no longer need to view/edit notes when setting them as “completed” or “dropped”.
You can still search completed notes from “Search All” pane.
Press Cmd+F/Ctrl+F again in the search bar to switch to Search All mode.
Or, as you said, please use tags instead for your note status management.