My funny little system of complex numbers

Hello my dear internet wanderer, it's me again, Lepersonaf. On this site i want to showcase you what i have designed recently. So, i was doing some complex math, like calculating some equations and all of this. At one moment i had to draw graphics on complex plane, and one of them caught my eye: it was this equation.

$${\displaystyle \begin{array}{c}\left|z\right|\le 1;\frac{\pi }{4}<\arg \left(z\right)<\frac{3\pi }{4}\end{array}}$$

I began to imagine how it will look like, and if i was correct, it should've be one quarter of a circle, rotated by 45 degrees. And amazingly i was correct! Here's how it looks.

After this i thought: "that's a nice graph, what if i can do mine?". And after one night of thinking and some trial and error, i came up with this formules:

$${\displaystyle \begin{array}{c}{z}_1:n\le \left|z_1\right|\le (n+1),n=\{0,2,4,...\};\text{⠀}0\le \arg \left(z_1\right)\le \frac{\pi }{2}\text{⠀}\text{⠀}\text{⠀}\end{array}}$$

$${\displaystyle \begin{array}{c}{z}_2:n\le \left|z_2\right|\le (n+1),n=\{1,3,5,...\};\text{⠀}\frac{\pi }{2}\le \arg \left(z_2\right)\le \pi \text{⠀}\text{⠀}\text{⠀}\end{array}}$$

$${\displaystyle \begin{array}{c}\text{⠀}{z}_3:n\le \left|z_3\right|\le (n+1),n=\{0,2,4,...\};\text{⠀}\pi \le \arg \left(z_3\right)\le \frac{3\pi }{2}\text{⠀}\text{⠀}\text{⠀}\end{array}}$$

$${\displaystyle \begin{array}{c}{z}_4:n\le \left|z_4\right|\le (n+1),n=\{1,3,5,...\};\text{⠀}\frac{3\pi }{2}\le \arg \left(z_4\right)\le 2\pi \text{⠀}\end{array}}$$

Two red-colored formules show two separate complex numbers, which are exactly the same, but have different angle rotation. Same with two blue ones. And as you can see from graph, it looks just amazing -- exactly as i imagined it:

Accroding to this formules, you get this beautiful ring pattern. I couldn't fit this function in one equation sadly, so i had to make a system. However, I'm still pleasantly surprised by result, because i didn't expect this to work out at all. GG's, and huge thanks to Copilot AI for helping me out with visualisation of this. Quite a fascinating system. Maybe i will do one more like this later...
Peace