Hi Ayush (and I see that Emily has made some small changes here as well)
I like this page, and it's totally new information to me, so I'm honestly better at telling you what's a bit confusing than exactly how to fix it...
The first thing that jumped to my mind as I started reading the sentence "Quaternions were discovered by William Rowan Hamilton on 16 October 1843." was, What was he doing when the discovered them? Why did he want to work with them? Do you have that information?
Here "Note, that even though v is called a vector, it is not a typical three dimensional vector but rather a vector in 4D space." I definitely get what you're saying, but I think two things would help explain it... turn that mentioning of a vector into a link to Vector and explain that there are three "complex" directions and one real one in the components of the vector. Can you give an example there, as well?
This equation: confuses me because it implies . Do you see why I'm confused? And what's going on here? I really want
In terms of the multiplication, it looks like you're doing something like a cross product. If that's true, you might explain similarities and differences with the cross product, because that might provide your readers with a link to something they understand more. You might also want to cite matrix multiplication as the canonical example of a non commutative multiplication.
Since you brought up the two identity quaternions, does that mean this is a field (in the algebra sense of the word)? If that question doesn't make sense to you, don't worry about it, but it might be cool to explore that a bit. In that section, why don't you work out an addition and a multiplication example to show that they don't change?
This: "When a quaternion is expressed as a 4x4 matrix" confused the hell out of me, since I haven't seen a matrix representation yet. Can you move that section down to below the bit about the matrix representation? I'd also keep your section about how it's used in computer graphics at the end of the page
Alright, that turned out to be a lot of things. BUT overall, I really did like this page. The content is really interesting!