the complex conjugate
The complex conjugate of z = x + i y is the number you get by flipping the sign of the imaginary part: z-bar = x - i y. Geometrically it is the mirror image of z across the real (horizontal) axis — a point above the axis reflects to one the same distance below. So conjugation is just reflection in the x-axis, the simplest symmetry the complex plane has.
Its magic is what happens when you multiply z by its own conjugate: z times z-bar = (x + i y)(x - i y) = x^2 - (i y)^2 = x^2 + y^2, a real, non-negative number — in fact exactly |z|^2, the squared distance from the origin. The imaginary parts always cancel. Conjugation also plays nicely with arithmetic: the conjugate of a sum is the sum of conjugates, and the conjugate of a product is the product of conjugates. And you recover the parts of z by Re z = (z + z-bar)/2, Im z = (z - z-bar)/(2i).
Conjugation is the workhorse trick for turning complex into real. It is exactly how you divide complex numbers (multiply top and bottom by the conjugate of the denominator to clear the i from below). It also captures a deep fact about polynomials with real coefficients: their roots come in conjugate pairs, so if 2 + 3i is a root then 2 - 3i is too. That symmetry threads through the whole subject.
For z = 3 + 4i, the conjugate is z-bar = 3 - 4i, and z times z-bar = 9 + 16 = 25, so |z| = 5.
Multiplying by the conjugate clears the i and leaves the real squared length |z|^2.
Conjugation reverses the sign of the argument (it reflects across the real axis) but leaves the modulus unchanged: |z-bar| = |z|. It is NOT a holomorphic operation — z-bar is the prototype function that fails the Cauchy-Riemann test, which is why it matters later.