That look is not that different. I bring attention to the fact that a product of quaternions all close to 1 can be close to -1, which correspond to a 2pi rotation too. This fact is a bit simpler to grasp than a topological explanation, where you consider a topology on the space of all rotations and show that some paths that correspond to 2pi rotation are different than the others. There are multiple youtube videos explaining the topological argument, see, e.g., this one https://www.youtube.com/watch?v=ACZC_XEyg9U.

The topological argument gives a larger picture and probably better understanding, but it is definitely harder.