Quotient relation on complex field

Question

How to make quotient relation on complex field, each equivalence class of which consist of only 2 elements?

Answer 1

If there are no other requirements than that every equivalence class has two elements, then there are many ways to go about it.

For example, we can define $x + iy \sim u + iv$ if and only if $y = v$ and $\lfloor \frac x2\rfloor = \lfloor \frac u2\rfloor$ and $x - \lfloor x\rfloor = u-\lfloor u\rfloor$. I.e. if the binary expansions of $x$ and $u$ are the same except for possibly the 1 bit.

As Robert Lewis suggests, the map carrying $z$ to its sole equivalent is not continuous in this example. Nor does it respect the field properties of $\Bbb C$. If you want an equivalence that does either, things become much tougher.