I have been dealing with complex numbers for few years now. But when I've tried to think about the motivation behind complex conjugation, I was not sure. Let me write what I am working with.
For a complex number $z \in\mathbb{C}$, where $z=\operatorname{Re}z+i\cdot \operatorname{Im}z$, we define complex conjugate of $z$ as $$ \overline{z} = \operatorname{Re}z-i\cdot \operatorname{Im}z. $$ Looking at complex numbers in the Gauss plane, this operation is symmetrical around the $x$-axis.
Question Is there any general motivation why we do that? (And after reading the rest of the question, is the motivation I've provided the right one, or are there others?)
I have studied linear algebra, so I know about involution, and adjoints/self-adjoints, where complex conjugation is a very nice example. My guess is that this comes from the fact about the roots of polynomials, where in the quadratic case, we have
$$ ax^2 + bx + c = 0 $$ and the solutions $$ x_{1,2} = \frac{-b\pm \sqrt{b^2-4ac}}{2a}. $$ And when $b^2-4ac < 0$, then $\sqrt{b^2-4ac}$ becomes imaginary \begin{align} \sqrt{(-1)\vert b^2-4ac\vert}=\sqrt{(-1)}\sqrt{\vert b^2-4ac\vert}=i\sqrt{\vert b^2-4ac\vert} \end{align} And we get the solutions $$ x_{1,2} = \frac{-b}{2a}\pm i\frac{\sqrt{\vert b^2-4ac\vert}}{2a} $$ which only differ in the sign before the imaginary part. Also in the general case, whenever $z$ is the root of $p$, then $\overline{z}$ is also root of $p$. Therefore creating the operation $\overline{\hphantom{a}\cdot\hphantom{a}}$ is justified.