Why do we not name $-a+bi$ in relation to $a+bi$? Given a complex number $a+bi$, it has a complex conjugate $a-bi$. The product of this complex number with its complex conjugate gives $(a+bi)(a-bi)=a^2+b^2$.
One might imagine flipping the sign of the real part instead of the imaginary part to get a sort of "anticonjugate", resulting in a similar product of $(a+bi)(-a+bi)=-(a^2+b^2)$.
Clearly this "anticonjugate" is the negative of the conjugate. I suspect I've never seen this concept before because of either (1) no finds this anticonjugate useful or (2) the negative of the complex conjugate is considered without giving it a special name.
Is there a different reason why we don't seem to use or consider "anticonjugates"? Or does the above account for this perception?

There is a tradition in math to avoid (or, at least, minimize) redundancies.
In this case, complex conjugation $\,\overline{a+ib}=a-ib\,$ has been known and used for a long time. It has many applications, from polynomial equations to calculus, abstract algebra, geometry etc.
In contrast, the proposed "anti-conjugate", say we write it as $\,\widetilde{a+ib}=-a+ib\,$, would be a new concept, without any obvious advantage - conceptual or practical. Moreover, it can be easily expressed in terms of the conjugate as $\,\widetilde{a+ib}=-\,\overline{a+ib}=\overline{i\cdot\overline{i \cdot(a+ib)}}\,$. Thus, redundant.

[ EDIT ] $\;$ Side by side summary.
$$
\begin{matrix}
 & & & \small{\text{conjugate}}\;\overline{\,z\,} & & & \small{\text{anti-conjugate}}\; \widetilde{\,z\,} 
\\ \small{\text{symmetry}} & & & \small{\text{over real axis}} & & & \small{\text{over imaginary axis}}
\\ \small{\text{involution}} & & \small{\text{yes:}} &  \overline{\overline{\,z\,}} = z & & \small{\text{yes:}} & \widetilde{\widetilde{\,z\,}} = z
\\ \small{\text{distrib over +}} & & \small{\text{yes:}} &  \overline{\,z_1+z_2\,} = \overline{\,z_1\,}+\overline{\,z_2\,} & & \small{\text{yes:}} & \widetilde{\,z_1+z_2\,} = \widetilde{\,z_1\,}+\overline{\,z_2\,}
\\ \small{\text{distrib over }\times} & & \small{\text{yes:}} &  \overline{\,z_1\,\cdot\,z_2\,} = \overline{\,z_1\,}\,\cdot\,\overline{\,z_2\,} & & \color{red}{\small{\text{no:}}} & \widetilde{\,z_1\,\cdot\,z_2\,} \ne \widetilde{\,z_1\,}\,\cdot\,\widetilde{\,z_2\,}
\end{matrix}
$$
