I've narrowed this down to only occuring when the worksheet in question is in
Page Layout View. Clicking some cells also selects the adjacent cell.
Changing the view to Normal restores normal behavior.

This only occurs on one machine. Another machine opening the same workbook
works fine in all views. I have taken the workbook in question and deleted
all data and the issue still happens.

If there is nothing obvious don't spend too much time on this as I have a
contact who might be able to help as well.