I apologize if this has been asked previously, but this seems to be a very particular question. For context, this is coming out of an attempt to define image pixel boundaries as a particular set of lines in the complex plane.
More precisely, I'm looking for a name/description of the following, which seems like a reasonable 'next step' from the Gaussian integers:
$\mathbf{B}[i] = \{{a+bi}$ | $a$ $\lor$ $b$ $\in\mathbf{Z}\}$
That is, this "Gaussian mesh' is just the set of all complex numbers which connect two neighboring Gaussian integers, or equivalently the set of all complex numbers where no more than one component is non-integer.
Does this object have a name? Does it have any salient properties?