To the issue you raise: What you call "my confusion" is based on my readings mainly of the papers of Rowan Hamilton, Maxwell, and the related Heaviside-Tait debates in Nature.
That was around 125 - 150 years ago. Isn't more recent experience relevant?
Vectors are absurdly easy to generalize. One can give them arbitrary sizes, and one can relate them to functions.
Vector = function of index variable
One can generalize vectors to tensors, vector-like objects that can have more than one index variable. They also are related to functions.
Quaternions? How do you generalize them?
One can get quaternions by the Cayley-Dickson construction. Start with the real numbers, then create a sequence of ordered pairs, with total size 1, 2, 4, 8, 16, 32, ... Quaternions are on the way in this sequence of "division algebras".
The real numbers are just a convenient algebraic field to use. The construction will work with any algebraic field that's the rational numbers or some superset of them.
Addition is component-by-component, and it keeps all the properties of the original addition operation. Multiplication and conjugation are nontrivial:
Definition: q = (q0, q1)
Conjugate: q* = (q0*, -q1)
Multiplication: q.r = (q0.r0 - r1*.q1, r1.q0 + q1.r0*)
Norm or length: ||q|| = q.q* = q*.q
Reciprocal = (conjugate)/(norm)
They are always distributive over addition. But as one continues one's construction, multiplication gets its properties changed.
0 1 Real numbers:
Multiplication: commutative, associative
Norms multiply: ||q||*||r|| = ||q.r|| = ||r.q||
Self-conjugate: q* = q
1 2 Complex numbers:
No longer self-conjugate
2 4 Quaternions:
No longer commutative
3 8 Octonions:
No longer associative. No matrix representation possible
Still alternative: associative with two variables equal
4 16 Sedenions:
No longer alternative
Still power-associative: order does not matter when constructing a power
Norms no longer multiply
There are pairs of nonzero elements that multiply to make zero
No changes for higher-order 2^n-ions.
How much use do they get in various applications?
Real numbers > complex numbers > quaternions > octonions > sedenions and higher