I understand the reasoning behind the idea that a beam of radioactive particles in motion will decay more slowly than one at rest due to time dilation. It's also easy to imagine how this is tested in an accelerator by comparing the decay rates from two beams of - say - muons, which are moving at different speeds.
It gets puzzling for me, however, when I try to imagine two observers moving at different speeds both measuring decay rates from the same beam of muons. Obviously this isn't possible, since in practice the two observers would never detect any of the same decays - each could only detect some of them, so neither observer gets an accurate measure of the decay rate. It seems odd though that the number of decays detected by a single observer is dependent on the velocity of that observer relative to the beam.
I imagine that the explanation falls into one of the following categories:
It's explained by relativity, and the fact that if I am moving along with the beam, it'll appear that the individual decays I detect have less energy than if the beam is moving very quickly, and this accounts somehow for the fact that I am detecting more decays. If so, how?
The explanation is actually just standard quantum weirdness. If so, can it be recast directly in terms of some other more familiar complementarity phenomenon?