I've lost count of the number of things I've painted/dyed/stained/coated purple, though I typically go for a more blue shade than that. To me, that looks like a pretty even 50/50 mix, perhaps slightly leaning towards the blue by a tiny amount, with the natural wood colour making up the difference. Say, 40% red to 60% blue. I find with wood, any purple mixture always ends up being more red/pink than you first think, hence why I always go for a bit more blue to compensate.
I believe that exact finish is what Agile/Rondo call 'tribal purple', which means the wood has actually been dyed black first, then sanded back before the purple dye is applied. You'll want to do this too, if you intend to replicate that look exactly.
Though bear in mind that most people do not have calibrated computer screens, so how they view any given colour will always be a bit off in unpredictable ways. I'm a photographer, so I have my screen calibrated for perfect colour and to me, on this screen, that looks like an fairly even mix of red and blue with just the wood tone pushing the final colour more towards the pink end. If your screen is more warm/cool-toned, it may be that the colour you're thinking of is not that. You also need to remember that every piece of wood will have a very slightly different colour to it, so you're going to end up having to experiment/guess no matter what, anyway. As I said, plain 50/50 (or nearly that) would be my guess, but YMMV.