As someone with absolute pitch, I should say I have not found it musically useful, and I would hesitate recommending people train for it even if it was possible. When I was learning to play jazz, it was actually a major crutch for me. For jazz especially, the modes, chords, progressions, and relationships between tones are really important and are what determines the 'color' of what you're playing. Harmony in general is determined by the relationships between tones, not the absolute tones themselves (mostly - there are of course differences in sound playing three octaves up or down), and an ear for absolute pitch doesn't help much there at all.
I could use absolute pitch to identify chords and modes, but it didn't help me 'think' about and play within them the way relative pitch does. A good ear for relative pitch, and an understanding of harmony go much farther.