Mathematica 10 introduced file name completion. It works with many functions, such as Import, ReadList, SetDirectory, etc., though there are some oversights: it doesn't work with SemanticImport and it doesn't work for paths relative to Directory[] if the current directory is changed.

Question: Is there a way to enable to enable file name completion for user defined functions?

Say, I have a function

importFormatX[file_] := ...

I would like file name completion to be triggered when I type importFormatX["....

Link to the SE version of this question.