Why do quaternions not only use two imaginary numbers. Can we not simplify quaternions
$$q = a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} \tag{1}$$
to the form
$$ \begin{align} q & = a + b\mathbf{i} + (c + d\mathbf{i})\mathbf{j} \tag{2}\\ & = e + f\mathbf{j} \end{align} $$ Where $e,f \in \mathbb{C}$.
Thus we only introduce one new imaginary number. Where we note that $\bf{ij} = k$, which holds true.
This would make quaternions seem more like complex numbers and we only need to define the interactions between $\bf i$ and $\bf j$.
The same would also hold true for octonions being represented as $s = g + h\mathbf{k'}$ where $g, h \in \mathbb{H}$. Where we see that we now have only 3 imaginary numbers instead of 7.
This 'factorized' representation seems to simplify things and make these numbers look more familiar.
This is already done. It uses constructions going back over 100 years.
For quaternions built from complex numbers, look up cyclic algebras. This leads to a central simple algebra using an arbitrary cyclic Galois extension, which for quaternions comes from the Galois extension $\mathbf C/\mathbf R$.
Note that $j$ does not commute with all of $\mathbf C$: $jz = \overline{z}j$. You have to decide if you want to treat the quaternions as a left $\mathbf C$-vector space or a right $\mathbf C$-vector space. As a vector space over $\mathbf R$, that subtlety does not arise. So a more compact representation brings its own new features.
The compact construction you describe for octonions as pairs of quaternions (as well as for sedenions as pairs of octonions) can be found in books or online under the name "Cayley-Dickson doubling process" or "Cayley-Dickson construction".