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We know that $ax+by=c$ with $gcd(a,b)=1$ could be solved over $\Bbb Z$. Supposing if $a,b,c\in\Bbb Z[i]$, is there an analogous framework to find $x,y\in\Bbb Z[i]$ (at least of minimum norms)?

Turbo
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    You can extend the Euclidean algorithm so to compute the Bezout identity exactly the same as for integers, e.g. as in this answer. Ditto for polynomials over a field or any other domain that has a (constructive) Euclidean algorithm. – Bill Dubuque Mar 31 '15 at 20:24

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