What is the best $1000 CD Player?

Just to complicate things a bit more, a good, flexible option for that much money might to combine a high-quality used DAC like the Ben Canto or Benchmark with a decent transport, perhaps a used Denon DVD player or Rotel CD player.

You'd get remarkable sound, an easy upgrade path and the possibility of experimenting with hard-drive based audio if that appeals to you.

Ditto Prima.

Used - Audio Aero Prima, Arcam fmj 23, Sony XA7ES.
New: Rega Appollo

I'm making an assumption that you appreciate the HDCD function of the Arcam. Look at a used Cary 300-200 or Resolution Audio CD-55.

ARCAM FMJ 23T if you have any HDCD discs. It is the primo HDCD player. If not, the AA Prima wins at that price.