Coprime Mappings on Gaussian Lines

If $A$ and $B$ are sets of integers, then a bijection $f: A\to B$ is called a {\it coprime mapping} if $a$ and $f(a)$ are coprime for all $a\in A$. The existence of coprime mappings between intervals of positive integers has been well studied. 
Here we extend this study to coprime mappings  between intervals of Gaussian integers on lines in the complex plane.  We give simple necessary and sufficient conditions that such  a line $L$ must satisfy to guarantee  that, for all $n\in\N$, a coprime mapping exists between any two contiguous intervals on $L$ consisting of $n$ Gaussian integers. For lines $L$ where these conditions are not met, we conjecture that there is a bound $B_L$ such that if $n>B_L$, then a coprime mapping exists between  any two contiguous intervals of length $n$  on $L$, and find such bounds for certain infinite families of lines. Finally, we consider coprime mappings on lines over other imaginary quadratic fields of class number one, and generalize the necessary and sufficient conditions for the existence of coprime mappings between contiguous intervals  to these lines.