Light Transport aircraft Captain reported another pilot on frequency was experiencing engine issues and was asked by ATC to write down a phone number while trying to immediately land. The Captain reported concerns with asking a pilot in distress to get 'superfluous information' while trying to focus during an emergency type situation.

Narrative

Aircraft on ZZZ TRACON frequency reported engine trouble and needed to land immediately; choosing ZZZ Airport. During the exchange with ATC they asked the pilot to write down a phone number to call if they landed safely. This is an unnecessary distraction during what could be a high stress life or death situation. It is a reminder to all of us that we need remember basics when it comes to flying safely: Aviate; Navigate; Communicate. I called ZZZ TRACON at the completion of our flight and spoke to a supervisor about this and Supervisor defended the position that it was appropriate. There are many documented instances of ATC detracting from the focus of aircrew during an emergency in order to get superfluous information. Discussion with regarding priorities during a flight emergency and not allowing ATC to distract from critical flight duties/procedures.This issue should be escalated with the FAA AND INCLUDE TRAINING FOR ATC CONTROLLERS SO THAT THEY UNDERSTAND THE GRAVITY OF DISTRACTING AIRCREW DURING AN EMERGENCY.