This was a pretty dry blog, focusing much more on the math behind quaternions than i intended. However, quaternions are still a fascinating topic that need to be understood for game development. It's hard, but not impossible to visualize quaternions. Also, quaternions tend to start making more sense when you start to think of them in terms of Geometric Algerbra
There are some other really good resources about quaternions that are pretty easy to find, check them out in case they make more sense. I haven't found any (game programming) books that cover quaternions in much depth, you might be stuck to online articles for this one.
When working with quaternions, i find this rotation converter and this visualizer to be extreamly valuable resources. I've also made my reference implementation for the code in this blog available under an MIT license. Non github link to reference code.