Is there a math concept or terminology for complex numbers with identical imaginary part and opposite real part? As we know, complex conjugation is the reflection symmetry of a complex with respect to the real axis. Is there a standard terminology for a pair of complex numbers which have identical imaginary parts and opposite real parts? If not, why such kinds of complex numbers are not as important as conjugate complex so that there is even no special math concept for them?
 A: The operation you are describing is simply $-\overline{z}$, so I'd say it would be overcomplicating things to introduce a new notation. The importance of complex conjugate over this other reflexion comes from the fact that $f(z) =\overline{z}$ is a field automorphism. Actually, it's the only non-trivial field automorphism of $\Bbb C$ that fixes $\Bbb R$.
A: One key point is that the conjugation map plays reasonably well with the algebraic structure: $\overline{x+y}=\overline{x}+\overline{y}$ and $\overline{x\cdot y}=\overline{x}\cdot \overline{y}$. Put another way, conjugation is an automorphism of the complex field. In fact, it's the only "reasonably nice" (e.g. continuous) nontrivial automorphism of $\mathbb{C}$ at all.
On the other hand, the map $$a+bi\mapsto (a+bi)^\star:=-a+bi$$ is not nearly as algebraically nice. It still plays well with addition, but not with multiplication: for example, $$1^\star\cdot1^\star=1\not=-1=(1\cdot 1)^\star.$$
A: For non-zero complex $z = re^{i\theta}, ~\overline{z} = re^{-i\theta}.$
For non-zero complex $z = x + iy = re^{i\theta}$, 
$w = -x + iy$ can be similarly expressed as  $re^{i(\pi - \theta)} = re^{-i\theta} \times e^{i\pi} = -\overline{z}.$
Alternatively, $(x - iy) \times -1 = (-x + iy).$
