Det. John Jones uses BT to Perplex up v Livewire's, (who is copiying PD) AV. Livewire then switched to Superman Ally to shoot at something Stealthy.
Is he still Perplexed?

Yes. He does not lose the Perplex.

'kay.
Thanks.

Just to clarify why, from the PAC

PERPLEX (optional): This effect ends immediately if this character (i.e. the one Perplexing) loses Perplex or is KOd, or if the target is damaged or healed during this turn.

BT only puts restrictions on when the Perplex may be used. Once it is in place, the normal rules for ending the effect come into play