Since the speed is infinite, the wavelength is always zero, no matter the frequency. But a wavelength of zero at infinite speed means an infinite number of waves hit the eye in any constant amount of time. Hence, and infinite amount of energy hits the eye at practically the same moment. But assuming your eye (and all of the universe) does not incinerate instantly, you will not be able to see the difference between red and blue, since they are perceived to have the same wavelength (zero). So yeah, technically you no longer have blue/redshifting.
The puzzling part is fun, because you are constantly learning new ways to use your body. See how to balance, how to move around, etc. I have found that dancing gives a similar learning challenge. Especially the more free-form dances like salsa and bachata. It’s fun learning new moves every week during the lesson, and then try to see if you can put them to practise during a party.
And don’t worry about beeing to stiff. If you can balance around boulders, you can get your body to move around for dancing too. Just takes some practise. I currently do both, and feel like I lack dexterity more for the climbing than for the dancing.
And unlike most of the hand-friendly options mentioned already, you do have to use your hands and arms a lot. Just not in a way that puts any stress on them.