This script deals with immortality. The entire cast all lived exceptionally long lives. Kevin McCarthy lived to be 96, Estelle Winwood was 101 when she passed away, Edgar Stehli passed away shortly after turning 89, and Dodie Heath died at the age of 96 in June of 2023
Coincidentally, the character portrayed by Kevin McCarthy in this episode mentions the date of September 11th. McCarthy died on September 11th, 2010.
The interior living room & front door entrance set from the first act is the living room of "George" in The Time Machine (1960). The original wood paneling from that film has been painted white, but is otherwise obviously the same set, albeit without the Victoriana dressings. The same set also appears in other first season episodes of The Twilight Zone.
The revolver Laurette uses to shoot Walter is a Colt 1862 (with an after-market cartridge conversion kit); an apt gun, for a man who fought in the Civil War, since it was first sold during the Civil War, and was a common side-arm, for both Union and Confederate soldiers.
A possible influence may be Mary Shelley's story, "The Mortal Immortal." It also deals with a man who receives the gift of immortality from a mysterious alchemist, but ultimately watches his loved ones die off.