Take a harmonic oscillator Hamiltonian like H(p,x) = p2 + x2. Then make the variable changes: a = p + ix and a+ = p - ix. Now your Hamiltonian becomes H(a,a+) = a+a or so. So a and a+ are just some other variables.
In QM this can be used, for instance for the QM harmonic oscillator, to construct eigenfunctions of the Hamiltonian by acting with these creation operators on the vacuum |0>.
In quantum field theory you normally promote the field itself to be an operator in order to treat space and time as labels of these fields; after all, in special relativity they are on equal foot so you shouldn't treat one as an operator (x) and the other as a label (t) parametrizing motion.
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