holomorphic as independent of z-bar
Here is the single most memorable slogan of the subject: a function is holomorphic exactly when it does not depend on z-bar. If you can write your function using only z (and constants) — never using z-bar, never using |z|, never using Re z or Im z except through z itself — then it is holomorphic. The moment a genuine z-bar sneaks in, holomorphy is lost.
Made precise with the Wirtinger derivative, this says df/dz-bar = 0. Since x = (z + z-bar)/2 and y = (z - z-bar)/(2i), any function of (x, y) can be rewritten as a function of z and z-bar; the slogan says holomorphic functions are exactly those in which the z-bar variable does not actually appear. So z^2, e^z, and 1/(z - 3) are holomorphic, while z-bar, |z|^2 = z times z-bar, Re z = (z + z-bar)/2, and z times z-bar^2 are not — each visibly contains z-bar.
This is a genuinely useful first test and a deep statement of WHY complex differentiability is so restrictive: a general smooth map of the plane can mix z and z-bar freely (it has two degrees of freedom, df/dz and df/dz-bar), but holomorphy throws half of them away by forcing df/dz-bar = 0. That single discarded direction is the source of all the rigidity — conformality, harmonicity, the identity theorem, everything. Just remember it is shorthand: z and z-bar are not literally independent, so 'no z-bar appears' really means the operator equation df/dz-bar = 0.
Quick test by inspection: f(z) = z^3 + 2z is holomorphic (no z-bar). g(z) = z + z-bar = 2 Re z = 2x is NOT (the z-bar is there, and indeed g maps everything to the real axis, so its derivative cannot be direction-independent).
Rewrite in z and z-bar; if z-bar genuinely appears, the function is not holomorphic.
It is a slogan, not a literal independence: z and z-bar determine each other. The rigorous content is df/dz-bar = 0, which is the Cauchy-Riemann equations.