On p. 87 of Dirac's Quantum Mechanics he introduces the quantum analog of the classical Poisson bracket1
[u,v] = ∑r(∂u∂qr∂u∂pr−∂u∂pr∂u∂qr)
as
uv−vu = i ℏ [u,v].
I'm not worried about the ℏ but if there is an (alternative) explanation of why the introduction of i is unavoidable that might help.
1 Note that Dirac uses square brackets to denote the Poisson bracket.
Answer
The imaginary unit i is there to turn quantum observables/selfadjoint operators into anti-selfadjoint operators, so that they form a Lie algebra wrt. the commutator.
Or equivalently, consider the Lie algebra of quantum observables/selfadjoint operators with the commutator divided with i as Lie bracket.
The latter Lie algebra corresponds in turn to the Poisson algebra of classical functions, cf. the correspondence principle.
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