Golden ratio $\phi$ and related numbers give the largest errors when approximated by rational numbers. I imagine that if we consider approximations of complex numbers by gaussian rationals or eisenstein rationals then other numbers may appear that are even more difficult to approximate.
My question is: is there a complex number $\xi$ such that there is at most finite number of gaussian rationals $a_n\over b_n$ such that $$ \left |\xi-\frac{a_n}{b_n}\right | < \frac{1}{\sqrt{5}\, |b_n|^2}$$ Another question: If yes, is there a real number $r$ such that for every complex $\xi$ there are infinite gaussian rationals $a_n\over b_n$ for which the following inequality holds? $$ \left |\xi-\frac{a_n}{b_n}\right | < \frac{1}{r\, |b_n|^2}.$$ Is there the largest such $r$? What numbers require the this largest $r$?
I also want to ask both of the questions above but with eisenstein rationals instead of gaussian rationals.