When describing sets of points in the complex plane such as $|2z-i| =4$ would it be correct to describe the set exactly as we would in the $x,y$ plane?
For instance, Letting $z=x+iy$ for some $x,y \in \mathbb{R}$
yields the equation $x^2+(y-\frac{1}{2})^2=4$
And would it be correct to say this is the circle in the complex plane described by $x^2+(y-\frac{1}{2})^2=4$
On
Would it be correct to describe the set exactly as we would in the x,y plane?
If there is no more structure given to $\mathbb{R}$ and $\mathbb{C}$ else than the norms, I will say yes, since we have the isomorphism $iso:\mathbb{C}\ni z=x+iy\mapsto (x,y)\in\mathbb{R}^2$ (bijection that conserving norm).
And would it be correct to say this ($|2z−i|=4$) is the circle in the complex plane described by $x^2+(y−\frac{1}{2})^2=4$
Yes, in sense that $$iso(\{z\in\mathbb{Z}:|2z-i|=4\})=\{(x,y)\in\mathbb{R}^2:x^2+(y-\frac{1}{2})^2=4\}.$$
I would say, rather, that this is the circle in the complex plane analogous to $\displaystyle x^2+\left(y-\frac{1}{2}\right)^2=4$.
The reason is, $\left |z-\frac{i}{2}\right|=2$ is well understood to mean a circle with radius $1$ and center $z=\frac{1}{2}$, and that the complex plane, as you note, uses $z$ instead of $x$ and $y$.
(Distance from center = radius)