A random question along those lines: why represent states as 2d complex vectors instead of quarterions? Aren't they the same thing? As soon as I read that I spent the rest of the article wondering if everything it would make even more sense cast that way.
Only in the sense that they are both require four real coefficients. The quaternions have a particular multiplicative structure that just doesn't apply to quantum states, so it doesn't make sense to use them.
That being said, the space of single-qubit operations is very much analogous to rotations in 3d and so is well described by quaternions. In fact, the Pauli matrices times i (iX,iY,iZ) are isomorphic to the quaternions (i,j,k). For example, iX * iY * iZ = -I.
If you have had more exposure to maths and computer science, it will be easier for you than someone with a "pure" physics background.
As for quarternions, yes they are isomorphic, but generally for useful applications, people consider quantum computers with n qubits. So your state is an element of C^(2^n). Apart from the measurement step, you can idealise any quantum computation as a unitary transformation, so an element of the unitary group U(2^n), acting on this complex vector.
An element of U(2^n) is representable as a 2^n x 2^n matrix U, with complex entries, st U.U^{\dagger} = I. Here dagger represents conjugate transpose, and I is the 2^n x 2^n identity matrix. Sometimes people add the extra constraint, det(U) = 1, then this gives you the special unitary group SU(2^n).
This is not to discourage anyone, but underselling it as requiring elementary linear algebra is not very helpful (the pop-sci articles have already been overselling it as "magical"/"mind-blowing" etc.).
For two qubits, the simplest entangled states are the Bell States[0] (generated from a CNOT and Hadamard gate). The article gives an example of one of them.
