I was proud of this idea, as it was probably the best early research idea I've come up with.

I became very interested in matrices and linear algebra after reading a paper on modelling origami using rotation and translation matrices (Belcastro and Hull, 2012). I began to play with expressing all kinds of things as matrices and seeing what "meanings" matrix operations had in those contexts.

Somehow complex numbers cropped up, and I decided that they were a good candidate for this "interpretation" because multiplying by a complex number meant a rotation and dilation of the complex plane - or an "amplitwist" (Needham, 1996). So I represented a complex number as a rotation matrix together with a scaling factor.

Given a complex number $z = r\mathrm{e}^{\mathrm{i}\theta}$, let $\mathbf{Z} = r\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}.$ Some interesting properties follow immediately from this correspondence:

Another old idea. Given a $2 \times 2$ matrix $\mathbf{A}$,

$$\det(\mathbf{A}) = \begin{bmatrix}1&0\end{bmatrix}\mathbf{A}^T\begin{bmatrix}0&1\\-1&0\end{bmatrix}\mathbf{A}\begin{bmatrix}0\\1\end{bmatrix}.$$
Prove this and extend it to general $n \times n$ matrices. Can you go further than that?