Multiple definitions for matrix representation of imaginary number

Question

I learnt that multiplying by i can be thought as 90 degree rotation since multiplying by -1 can be thought as 180 degree rotation.
$i*i=-1$ and
$ 90^\circ rotation(90^\circ rotation(x))= 180^\circ rotation(x)$
90 degree rotation require a new plane so invent a complex plane where base unit is i since base can be created by $1\times i$
This idea reminded me of linear transformation.
where $90^ \circ$ rotation(x) = $\begin{bmatrix}0&-1 \\ 1&0 \end{bmatrix}\cdot x$
and
$180^ \circ$ rotation(x) = $\begin{bmatrix}-1&0\\0&-1\end{bmatrix}\cdot x$
So I thought I could just define
$i=\begin{bmatrix}0&-1 \\ 1&0 \end{bmatrix}$and $-1 =\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$
This idea seems to work for the most part
$i \times i = -1, i+i=2i$
But then, I would have to define
$i = \begin{bmatrix}0\\1\end{bmatrix}$and$-1 = \begin{bmatrix}-1\\0\end{bmatrix}$ in order to treat two vector as a complex number.

This feels like I am doing something wrong since I have two seporate definitions for what is suppused to be the same thing and to make things worse I have to use different definition for i when I am using different operation on them.
E.g.
$\begin{bmatrix}0&-1 \\ 1&0 \end{bmatrix}\cdot \begin{bmatrix}0\\1\end{bmatrix}=\begin{bmatrix}-1\\0\end{bmatrix}=-1$ but
$\begin{bmatrix}0&-1 \\ 1&0 \end{bmatrix}\cdot \begin{bmatrix}0\\1\end{bmatrix} \neq \begin{bmatrix}-1&0\\0&-1\end{bmatrix}=-1$
Here I have a contradiction which depends on which definition I use.
And there are other cases where things just don't make sense in terms of linear algebra e.g.
$\begin{bmatrix}0\\1\end{bmatrix}\cdot\begin{bmatrix}0&-1 \\ 1&0 \end{bmatrix}=\begin{bmatrix}0\\-1\end{bmatrix}=-i$? but
So it feels wrong to represent imaginary number in form of matrices but I don't really know where this confusion comes from.
It might be a bit of an open ended question but I would appreciate any help.

Answer 1

There is actually something called the matrix representation of a complex number. $$ a + bi = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$$ You can show that multiplying and adding these matrices works will represent the same complex number that we would get by multiplying and adding the complex numbers that the matrices represent. This representation of a complex number is exactly what you derived for your first definition.

As to tackle the two definitions problem, think of the matrices as operations that multiply your vector representation of $a + bi $ as $\begin{bmatrix} a \\ b \end{bmatrix} $. The matrix you defined to be $i$ is actually just an operator that will multiply your vector by $i$, and the matrix you defined to be $-1$ is just an operator that multiplies your vector by $-1$.

For more on the matrix representation of the complex numbers with its derivation: https://the-educational-blog.quora.com/Matrix-representation-of-complex-numbers

Hope this clears up some confusion.