We know that for any complex number $z = x + \iota y$, where $x$ and $y$ are real numbers, there exists the complex number $\overline{z} = x - \iota y$, and the complex numbers $z$ and $\overline{z}$ are said to be the complex conjugates of each other.
Of course, every real number is its own conjugate.
Now is there any generalization of this notion of conjugate pairs to elements of a general abstract field?
The complex numbers are an extension of the real numbers, and complex conjugation is a field automorphism of the complex numbers fixing the reals.
In general, if $F$ is a field and $E$ is an extension of $F$ (that is, a field containing $F$), then one can speak of the automorphisms of $E$ that fix the elements of $F$. That's the idea analogous to complex conjugate. It's a really, really important concept. It forms the basis of Galois Theory, for one thing.