Is there a way to automatically add another member to a chat each time a chat is started with a particular member? 

e.g. a member starts a private chat with the Administrator and a particular Moderator is automatically added in. 

Doable via plugin. PM sent. 🙂

There is no way in which to do this without customisation, so indeed it would be a plugin required for something like this

And will be done. Job taken. 👍