non unique imaginary units A professor told me that another way to define complex numbers is through the real matrices of the form  $\begin{bmatrix}a & b\\-b &a\end{bmatrix}$. With this definition, the imaginary unit becomes the matrix $I=\begin{bmatrix}0 &1\\-1&0\end{bmatrix}$. By definition $i^2=-1$ and It's pretty easy to check that the square of the matrix $I$ is the negative of the identity matrix.
So apparently, any matrix with square equal to the negative of the identity can be taken as $I$. Why is that? When you think about the $2D$ plane, this corresponds to choosing a vector different from (1,0) as $\bar i$, the unit vector in the x-direction.
Then he says that by some general theorem all complex structures in dimension 2 are isomorphic to each other and that one can allow the imaginary unit to change from point to point.
Does anyone understand what this means and why we can do this?

$(0,1)$ and $(0,-1)$ are equally good to be taken as square root of $-1$. The isomorphism that is being talked about is the map $(x,y) \to (x, -y)$.
Incidentally, the definition of $i$ as $\sqrt {-1}$ found in some books is quite meaningless.
