Say I want to find the order of an element in $\mathbb Z[i]$ modulo $a+bi.$ If this element is $x+yi,$ I could write $(a+bi)(c+di)+1=(x+yi)^n.$ I noticed that if $y=0,$ we could do this pretty easily by considering the order of $x$ modulo $N(a+bi)=a^2+b^2,$ but for nonreal or nonimaginary elements this seems harder and maybe needs more computation.
How can I calculate the order $n$? Is it entirely algebraic?