I recently upgraded to Mojave from 10.9. The app store shows an old Apple id as the "owner" of iMovie and I cannot update my version of iMovie without a password I don't know. My old version of iMovie works on Mojave and did on Mavericks however. I don't remember paying for iMovie. The App Store reports Apr 2014 as the last update.
How do I update iMovie now that I'm running macOS Mojave and using a different Apple ID?
Best Answer
iMovie and the other iLife apps used to be paid-for except with a new Mac, and when the App Store was introduced, it registered the Mac's serial number with your App Store account to allow you to download iMovie etc. for free.
These days, these apps are actually free anyway. The problem in your case is that you still have the old binary around, and this contains the App Store receipt with your old App Store Apple ID.
Step 3 is crucial, if the App Store can find an existing copy of the app it will try to update it again, running into the same problem as before.