the difference quotient
Before you can take a derivative you need an honest 'average rate of change' over a small step — that average is the difference quotient. For a complex function it is the single expression (f(z) - f(z_0)) / (z - z_0): the change in output divided by the change in input, both measured as complex numbers.
Read this carefully as a complex division, not two separate real fractions. The numerator f(z) - f(z_0) is a complex displacement (it has a length and a direction), the denominator z - z_0 is another complex displacement, and dividing them rotates and rescales the first by the second. As z slides toward z_0 the step shrinks; if the quotient settles to a definite complex number regardless of the direction of approach, that number is the complex derivative f'(z_0).
The difference quotient is where the rigidity of complex analysis first bites. Along the real axis the step z - z_0 is real; along the imaginary axis it is purely imaginary, so dividing by it includes a hidden factor of 1/i = -i. Insisting that both routes give the SAME limit is exactly what forces the Cauchy-Riemann equations — so this humble fraction already contains the seed of the whole theory.
For f(z) = z-bar (the conjugate), approach along the real axis: the quotient is 1; along the imaginary axis it is -1. The two disagree, so the difference quotient has no limit and f'(z) does not exist anywhere.
Conjugation is a clean example of a smooth-looking map whose difference quotient depends on direction.
It is a complex fraction, not two real ones glued together — that single division is what carries both a scaling and a rotation, and what the limit must respect.