Step 1: Link the concept to the person. The question asks who mapped out how deep a patient goes under ether. That graded description of anaesthetic depth, built on observable bedside signs, was the work of Guedel, whose name is permanently attached to these stages.
Step 2: Outline what he described. Guedel divided the course of ether anaesthesia into four sequential stages running from light analgesia, through an excitatory phase, into surgical anaesthesia with its planes, and finally to dangerous medullary depression. Pupillary changes, reflex loss, and breathing changes guided the staging.
Step 3: Reject the other names. Morton is remembered for the historic 1846 public ether demonstration but did not create the staging system, and Thompson has no role here. So the correct attribution for the levels of ether anaesthesia is Guedel.
\[\boxed{\text{Guedel (option 2)}}\]