In the case of $\Bbb N$ and $\Bbb Z$ the gap between two consecutive primes could be defined roughly speaking as the absolute value of the (1-dimensional) distance between those mentioned consecutive prime numbers, but is there an equivalent for the Gaussian primes in the complex plane?
For instance, this is just an example, I was thinking about a distance defined by the polar coordinates, in two variables, $(r,\alpha)$, being $r$ the Euclidean norm of the complex number and $\alpha$ the angle between the line that joins $0+0i$ and the Gaussian prime $a+bi$ and the x-axis ($\Bbb R$). A smaller angle would be a smaller number, so in the case of two Gaussian primes with the same distance to $0+0i$ (belonging to a same circunference of complex numbers centered on $0+0i$) the smallest one would be the closest prime to $\Bbb R$ (in some cases it would be a prime $\in \Bbb R$ when possible according to the definition of the Gaussian primes). So it would be a two-dimensional distance (probably it would be possible to transform it into a one-dimensional distance).
Regarding the gap between Gaussian primes I found this article: "Bounded gaps between Gaussian primes" but it seems to focus on gaps between two specific primes, but not considering a complete order between all of them. Also it seemed interesting this presentation about if there are infinitely large gaps between Gaussian primes
I would like to ask the following questions:
How is defined the gap between Gaussian primes?
Is there a standard way of defining a total order between Gaussian primes (or in the complex plane $\Bbb C$ in general)?
If somebody could kindly provide some online articles for some orientation on those topics would be very appreciated. Thank you!