Complex structure through two involutions The standard algebraic definition of a complex structure $I$ is $I^2=-1$. On real pairs $(a,b)$ it is represented like $(a,b)\mapsto (-b,a)$. But what if we have not had the negative pairs of reals $-(a,b)=(-a,-b)\quad\Leftrightarrow\quad I^2=-1$  and had only the single real involution $a\mapsto -a$. Is it known a definition of $I$ through composition of two independent involutions on couples? I mean the swap $(a,b)\mapsto (b,a)$ and the one entry sign change $(a,b)\mapsto (-a,b)$? They both look more primitive so more fundamental.

I'm not sure exactly what you're asking. On the surface, it looks like you're excited about the factorization $\begin{bmatrix}-1&0\\0&1\end{bmatrix}\begin{bmatrix}0&1\\1&0\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}=\begin{bmatrix}-b\\a\end{bmatrix}$.
Of course, there is an equally plausible decomposition using complex conjugation rather than negation of the real numbers $(a,b)\mapsto (a,-b)$ which would yield a factorization 
$\begin{bmatrix}0&1\\1&0\end{bmatrix}\begin{bmatrix}1&0\\0&-1\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}=\begin{bmatrix}-b\\a\end{bmatrix}$.
We can play these games all day with different involutions. The question is: are these that look special(/fundamental/primitive) to us give us any insights?
Personally I don't see any. The "specialness" as far as I can see is just an artifact of the basis we are working with. The product of these particular involution is not fundamentally different from the product of any two other involutions.
