Is it possible to jump to a particular date in the past (say January 6, 2013) to see the completed tasks that day?
Unfortunately, this is not possible at this time. You can click on the Karma link at the top and then "view all completed" and you are able to sort by date, but you can not search for a particular date itself.
We will consider improving this in future updates. Feel free to add this to our votebox: http://todoist.com/Vote/
Thanks for the quick reply. I added a recommendation for this in the Votebox ("Add search capability to jump to a particular date in the past to see the completed tasks that day.")
I like to be able to be able to check past actions & dates.