Complex number ABC Conjecture Regarding the abc conjecture, I can't find records for Gaussian integers.
Let $rad(a) = \lVert{\prod{prime factors(a)}}\rVert$.
For relatively prime $(a,b,c)$ with $a+b=c$, define quality as 
$Q = \frac{\log{c}}{\log(rad(a b c))}$.  
For Gaussian integers, the highest quality I can find in initial searching is 
$Q = 1.65169:\ -1 +\ (2 + i)^4 =\ i(1+2 i)(1+i)^7 :  $
$Q = 1.50515:\ 1 +\ (2 + i)^2 =\ -(1+i)^5 :  $
$Q = 1.34527:\ 1 +\ (8 + 7 i)^2 =\ -i(1+2 i)^2(1+i)^9 :  $ 
$Q = 1.23754:\ -3^4 +\ (4 + i)^4 =\ -i(1+2 i)(2+ i)^2(1+i)^9 :  $
$Q = 1.21850:\ (1+2 i)^2 +\ (4+i)^4  =\ -(1+i)^2 (3+2 i)^2 (2+i)^3 : $
$Q = 1.21684:\ -1 +\ (4+i)^4  =\ -i(1+i)^8(1+2 i)(2+i)(2+3 i) : $
Can anyone beat that first item, or supply other complex a+b=c sums with $Q>1.22$?

For equation $a+b=c$ $$i(1+i)\;+\;(2+3i)^4\;=\;i(3+2i)^4$$
we have 
$$Q = \frac{\log\left(13^2\right)}{\log\left(13\sqrt{2}\right)}\approx 1.761929.$$
