The imaginary part of

$3+2 \sqrt{-54}=3+6 \sqrt{6} i$

Let $\sqrt{3+6 \sqrt{6} i}=a+i b$

$\Rightarrow a^{2}-b^{2}=3$ and $a b=3 \sqrt{6}$

$\Rightarrow a^{2}+b^{2}=\sqrt{\left(a^{2}-b^{2}\right)^{2}+4 a^{2} b^{2}}=15$

So, $a=\pm 3$ and $b=\pm \sqrt{6}$

$\sqrt{3+6 \sqrt{6} i}=\pm(3+\sqrt{6} i)$

Similarly, $\sqrt{3-6 \sqrt{6} i}=\pm(3-\sqrt{6} i)$

$\operatorname{lm}(\sqrt{3+6 \sqrt{6} i}-\sqrt{3-6 \sqrt{6} i})=\pm 2 \sqrt{6}$