In general, we invent definitions and notation because we intend to make use of them somewhere. For example, the term "absolute value" and the corresponding notation $|-|$ exist because we regularly have occasion to refer to it in e.g. the definitions of variance of a random variable, limit of a sequence, and other constructions. For every one of these applications, the corresponding concept in the complex numbers is captured by magnitude. In contrast, I cannot think of a single case where I have needed to refer to "the number in the first quadrant differing from this one by a factor of a power of $i$."
Moreover, the useful algebraic properties of the absolute value function on the reals are not true of the function you've described. For example, if we denote your function by $[-]$, it is not the case in general that $[xy] = [x][y]$ -- how could it be, since the first quadrant isn't even closed under multiplication?
The point is, you can define whatever function you want, but if it's just a curiosity and not something that comes up naturally then it's probably not worth endowing with its own special notation and terminology.