Related to the construction of $\Bbb C$ (generalisation) To construct $\Bbb C$, we consider $\Bbb R^2$ endowed with the operations: $$\begin{align} (a,b) + (c,d) &:= (a+c, b+d) \\ (a,b) \cdot (c,d) &:= (ac - bd, ad+bc)\end{align} $$
then write $(0,1_{\Bbb R}) = i$, go on writing $(a,b)$ as $a+ib$, etc. Maybe the question is silly and has a trivial explanation, but nevertheless, I'll ask: Has anyone ever tried to repeat the procedure with $\Bbb C^2$, "nesting imaginary units"?
More specifically, I mean, define on $\Bbb C^2$ the operations: $$\begin{align} (z_1,z_2) + (w_1,w_2) &:= (z_1 + w_1, z_2+w_2) \\ (z_1,z_2) \cdot (w_1,w_2) &:= (z_1w_1 - z_2w_2, z_1w_2+z_2w_1)\end{align}$$
then write $(0,1_{\Bbb C}) = j$, where $j$ is another "imaginary unit" such that $j^2 = -1_{\Bbb C}$, write $(z_1,z_2) = z_1 + jz_2$ and so on? (I'm just using this subscript $\Bbb C$ for emphasis). It seems to me that we would get $i^2 = j^2$, but that wouldn't mean necessarily that $i = j$, would it? Also, would be this related in any way to the quaternions?
This, way, if we can make $\Bbb C^2$ a field, we could make any $\Bbb R^{2n}, n \in \Bbb Z$ a field, by repeating the process, no? I don't know if any problem would appear if I expanded everything in terms of the real and imaginary parts of every component. We would get some mixed terms $ij$ and $ji$. Surely I could adventure myself in the calculations, but if someone already thought of this, and it ended up being meaningless, I won't keep hitting my head on the wall. Thank you for the attention, and I hope I managed to get my idea through.

Close. To get the complexes, use real matrices
$$ 
 \left(  \begin{array}{rr}
  a  &  b  \\
   -b   &  a  
\end{array} 
  \right)  .
  $$
To get the quaternions, use complex matrices
$$ 
 \left(  \begin{array}{rr}
  \alpha  &  \beta  \\
   -\bar{\beta}   &  \bar{\alpha}  
\end{array} 
  \right)  .
  $$
