This is not a question but a solution. Perhaps everyone knows this already, but it took my some time to figure out so I thought I’d share it.
I am using jupyter lab 4 with julia (julia 1.9.2, IJulia 1.24.2). In notebook cells, tab completion for unicode characters like \mu works fine. However, it does not work out of the box in the text editor, which is highly annoying. To solve this, you have to open a console for the editor: Right click in the editor window and “Create Console for Editor”. Tab completion for unicode then works. You can even close the console and it continues to works.