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It seems that $i$ plays an important role in quantum mechanics (Q.M.). On the other hand, linear algebra plays such an important role in Q.M. too. So would linear algebra, such as a matrix be able to represent the formalism of Q.M. totally?
$i$ is the symmetric decoupling of $-1$, and in matrix representation, the $i$ is also a requisite for Hamiltonian symmetric decoupling of the negative identity matrix.
On the other hand, $i$ could be viewed as a phase factor. Would it be possible to replace $i$ by a $2\times 2$ matrix?