When Beckett questions the reasoning of why Castle wants her to believe in Mediums etc. he replies with: '... if you don't believe in even the possibility of magic you'll never ever find it.' He is paraphrasing a famous quote by Roald Dahl ('Those who don't believe in magic will never find it.')
Castle reveals to Beckett his given name is Richard Alexander Rogers. He changed his name professionally to Richard Edgar Castle, choosing Edgar in honor of Edgar Allen Poe.
Castle calls Beckett 'Scully', the cynical agent from The X-Files (1993). Castle (2009) executive producer David Amann was also a producer on The X-Files (1993).
The 'role' of the dead psychic is played by Patricia Tallman, the actress perhaps best known for her role as Lyta Alexander on the series Babylon 5 (1993). The Lyta Alexander character is a very powerful psychic.