I need to trigger a process builder when an approval process has approved a record. Instead of explicitly updating a field, is there a standard field update that happens with approval? (For which I can trigger the process)
The easiest way to handle your scenario is to add a field to whatever object you're approving and have your final approval step be updating that field. Then, have your process kick off of that field update.
This field doesn't need to be complex, for instance a checkbox field called "Trigger Process" would work or having a picklist field with "Accepted", "Rejected", and "In Progress" with "In Progress" set as the default then "Accepted" as the final action for acceptance state and "Rejected" in the rejected state.
Most of the action you can take from approval processes relate to workflow rules, such as: tasks, outbound messages, etc. Unfortunately, there is currently not a direct way to trigger Process Builder from Approvals that I know of.