What is the congruence class of a complex number modulo a prime?

Question

If we are given a complex number $a+bi$, modulo a prime $p$, in the form of the two integers $a$ and $b$, can we say which fractions in the form $\frac{c_1}{c_2} + \frac{d_1}{d_2}i$ are equivalent to this value? In other words, we have $a$, $b$, and $p$, and I want to find all fractions that are equivalent to $a+bi$ modulo $p$.

Answer 1

You would like to have $\frac{c1}{c_2}\equiv c_1 c_2^{-1}\equiv a \pmod p$. If you choose $c_1$ you compute $c_2\equiv c_1 a^{-1} \pmod p $ and if you specify $c_2$ you simply compute $c_1 = c_2 a \pmod p$. Same procedure for $d_1,d_2, b$