<s>[INST] Quadratic sum of Gauss integers Let $A$ be a set, define $nA=\{x\mid x=a_1+a_2+\cdots a_k,a_i\in A,1\leq k\leq n\}$.
Denote $G=\{z\mid z=(a+bI)^2,a,b\in \mathbb Z,I=\sqrt{-1}\},K=\{z\mid z=a+2bI,a,b\in \mathbb Z,I=\sqrt{-1}\}$

What's the smallest integer $n$ such that $K=nG$ (if there exist)?

Lagrange's four-square theorem states that any natural number can be represented as the sum of four integer squares: $a=x_1^2+x_2^2+x_3^2+x_4^2,$ hence $a+2bI=x_1^2+x_2^2+x_3^2+x_4^2+b\cdot(1+I)^2$.
$a+bI~(2\not\mid b)$ cannot be represented as the sum of several integer squares, because $2\mid \Im(x+yI)^2=2xy$.
 [/INST] (This is not a full answer, but too long for a comment)

Claim: $n\leq 8$.

Given $a+2bi\in\mathbb Z[i]$ for some $a,b\in\mathbb Z$. By Langrange's theorem, we have:
$$|a|=x_1^2+x_2^2+x_3^2+x_4^2$$
$$|b|=y_1^2+y_2^2+y_3^2+y_4^2$$
for some $x_i,y_i\in\mathbb Z$. Now, define
$$c_a=\begin{cases}1&a\geq 0\\i&a<0\end{cases}\qquad c_b=\begin{cases}1+i&b\geq 0\\1-i&b<0\end{cases}$$
Then
$$\begin{align}
&\quad (c_ax_1)^2+(c_ax_2)^2+(c_ax_3)^2+(c_ax_4)^2+(c_by_1)^2+(c_by_2)^2+(c_by_3)^2+(c_by_4)^2 \\
&=c_a^2\left(x_1^2+x_2^2+x_3^2+x_4^2\right) + c_b^2\cdot\left(y_1^2+y_2^2+y_3^2+y_4^2\right) \\
&=c_a^2\cdot |a| + c_b^2\cdot |b| \\
&=a+2bi
\end{align}$$
(Check, that the last equation holds for all four cases of $(c_a,c_b)$.)
</s>