As given in the book by Pollard, Diamond, on Algebraic Numbers, there is proof for division algorithm being made to work for Gaussian integers, with the above as an exercise left to the reader to prove by an example, as shown below in pg. 8,9 of the book.
But, I am unable to theoretically imagine the significance (say, by relation of lattice points of one set of the two values, to another set) and the cause of non-uniqueness being there in Gaussian integers (and hence in all algebraic integers), & so feel that an example would be understandable or workable by me, once I can derive the proof of the same.
However, there is an example copied from example at page 6 of this, that states example of dividend of $3+2i$, divisor of $-1+3i$, and states that as long as the norm of the remainder is less than that of the norm of divisor. The example has dividend as : $3 + 2i$ with norm = $13$, divisor as : $-1 + 3i$ with norm = $10$, and multiplying the fraction with conjugate of the denominator(divisor) yielding : $\frac{3}{10} + i\frac{-11}{10}$.
The first set of values derived is : $q = -i, r = i$. There is derivation of the first set of values by a similar mechanism to that stated in the book, but it is not clear that how the ratio of $3/10$ being closer to $0$, & of $-11/10$ being closer to $-1$ leads to quotient of $-i$.
The alternate value set suggested is: $q = 1-i$, $r= 1-2i$, as the norm of remainder $\lt$ than that of divisor.
$(-1 + 3i)(1-i) + (1-2i) \implies (-1 +4i +4) + (1 -2i) = 3 +2i$. There is no clue to me how to check for different such quotients. Can I take any quotient with $N_q \lt N_b$, ($b$ standing for divisor), and being a rational expression will succeed. So, the first step should be the check of dividend being not a prime. But, how? I mean the check should be $(a+bi)(x+yi) = 3 + 2i$, and its conjugate $(a-bi)(x-yi) = 3 - 2i, \exists a,b,x,y \in \mathbb {Z}$. This should lead to :$$
\begin{align}
& (a+bi)(x+yi) = 3 + 2i & \ (a-bi)(x-yi) = 3 - 2i \\
& ax - by =3, ay + bx =2 & \ ax -by =3, ay + bx = 2 \\
\end{align}
$$ On multiplying the two conjugates on both the $lhs$ and $rhs$ lead to :
$(a^2 + b^2)(x^2 + y^2) =13$ which to me leads nowhere, as the dividend should not be a prime, but cannot find two non-unit factors of $13$.
Please tell me where I am wrong, as it is not possible to consider the ratio of $a/b$ to be considered for checking primality. So, what is wrong? Totally confused!
Also, as earlier stated without a proof, or understanding the significance in terms of lattice points, it is banal to find solutions for quotient and remainder.