The simple way to access the timestamps for when a status was last changed is to go to an EventLog page and create a filter like below that is searching for the specific Status change to cmpt. The one thing that this wouldn’t allow for is if someone changed a Status to cmpt back to ip then back to cmpt again, that task status change would get counted twice. Hopefully, that doesn’t happen too often but in the world of production, anything is possible. You can use a filter system like this for any entity status change, you would just have to tweak the filter a bit.
Result:
The most effective but more involved solution would be to leverage the EventLogEntries with the API. I’ve seen users that have created a complimentary date field to the Status field that is populated via the API with the date of the most recent Status change. That way you can filter for Status is “Complete” and Status Update is “Today” (or any date timeline) to pull the data you like (whereby Status Update is the new field you create).