Describing points in the complex plane 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 $x^2+(y-\frac{1}{2})^2=4$


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 $$iso(\{z\in\mathbb{Z}:|2z-i|=4\})=\{(x,y)\in\mathbb{R}^2:x^2+(y-\frac{1}{2})^2=4\}.$$
