When you say the Completion Date and Completed By, are you referring to when the "Complete" action is performed on the task? If so, this will be creating a "Completion" record which is a collection on the task, and the Creation Date and Creation User attributes on that Completion record will be the data you need.
To get that data onto the tab at the bottom of your Change window you'll need to copy it to the task itself. To do that create two new attributes on your Task object and you can use calculations to populate them when the Completion record is created.
This would be very similar to the Current Assignment Group and Current Assignment Analyst calculation attributes on the Incident object, which is documented here: The Current Assignment Group and Analyst attributes show no or old data, or for more general information on how to copy the values see this document: How to automatically set or update values on an incident.
I hope that helps.