I play instruments variably at modern and baroque pitch, so am very glad that I don't have absolute pitch: it'd be a total pain. It's bad enough trying to learn to play instruments which have the same sets of fingerings to denote different notes: I'm still only a little way into learning to play my voice flute, in D, when normally I play recorders either in F or in C. Even learning those two sets was complicated by having to learn them in bass and treble clef.
When I came down to Egham, I handed the voice flute over to one of the forums members who has absolute pitch.... and I have a feeling the experience of playing that may have been just a little traumatic for her, as it's in baroque pitch as well as in D.
Yesterday I was being accompanied by another forums member who has absolute pitch: he had a full strings score from which to collate his accompaniment, we were again playing in baroque pitch, and he was putting together most of his part from the viola line, ie from alto clef

. A more remarkable achievement as he hasn't formally learned to play from that, and couldn't use his absolute pitch to help. He tells me that he can "shift" his pitch perception a little after a while, thanks to having listened to a lot of early music, but hates "playing between the cracks" - so a'=415Hz is okay because that is a semitone out of his reference range, but quarter tones hurt.