The primes of the form $4n^2-8n-9$ are just those of the form $n^2-13.$ Various substitutions can change this into many other forms. Is there a canonical choice for the polynomial, such that $C(p_1)=C(p_2)$ iff the primes of the form $p_1$ are those of the form $p_2$? (I suppose such form must exist; I guess the real question is whether there is a canonical canonical form.
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If the polynomials are quadratic, then there's a whole theory of equivalence classes of quadratic forms and the canonical member of any class is the one that is "reduced." The definition of this technical term should be available wherever quadratic forms are discussed.
I don't know about reduced forms in higher degree.