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?
 A: In a similar way, we can represent the real number $-1$ by the matrix: $\begin{bmatrix}-1 &0\\0&-1\end{bmatrix}$ which can also be thought of as rotation by $180^\circ$.  Is it rotation clockwise or anticlockwise?  Well for $180^\circ$ it doesn't matter.
However, $i$ represents rotation by $90^\circ$ and clockwise and anticlockwise make a difference.  Which is right?  It doesn't matter, just make a choice and stick to it.
Compare driving on the road.  Should we drive on the left or the right?  Clearly both work as some countries choose one and some choose the other.  However, it is important for a country to make a choice and be consistent.  Also, when visiting a country with the opposite convention, care is required to adjust.  The UK and France have similar road rules except that left and right are flipped.  For example, when joining a roundabout in the UK you give way (yield) to traffic already on the roundabout, this traffic is approaching from your right.  In France, you also give way to traffic already on the roundabout but it is approaching from your left.
Please note that this is only a brief description of the differences between driving in the UK and France.  If you actually plan to drive in France, please research the rules more thoroughly.  Swapping left and right is just a first approximation.
A: $(0,1)$ 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.
