linear algebra - Different interpretations of imaginary number

I went through a linear algebra course and I'm a bit confused..

I think I understand the geometric interpretation of imaginary numbers - multiplying by $i$ results in rotation by $90$ degrees in so that $1$ becomes $i$ and so forth. And this is where that $i^2 = -1$ comes from.

And then there's the matrix representation of $i$, which I understand emerged from a later generalization of complex numbers. I interpret the matrix representation as transform function which basically projects the imaginary axis to the real axis. I've thought of it as something very similar to vectors, with the difference that with vectors I write:

$P = x\mathbf{\hat{i}} + y\mathbf{\hat{j}}$ where $\mathbf{\hat{i}} = (1, 0)$ and $\mathbf{\hat{j}} = (0, 1)$

..and with complex numbers I can write:

$C = a + bi$ where $i$ = $2\times 2$ matrix, which represents the same $90$ degree transform logic by transformation.

Correct? Or at least close?

Anyways, as I understand, both of these interpretations of $i$ are actually later than $i=\sqrt{-1}$ itself. Is there an earlier interpretation? How did those who invented imaginary number prove that $i = \sqrt{-1}$ in the first place?

Thanks!