writing f as u + i v
A complex output w = f(z) is itself a complex number, so it splits into a real part and an imaginary part. Writing z = x + i y for the input and f(z) = u + i v for the output, the two real numbers u and v each depend on the two real numbers x and y. So a single complex function f secretly carries two real functions of two real variables: u(x, y) and v(x, y). We write f = u + i v.
Unpacking f this way turns one complex equation into two real ones, which is exactly how you connect complex analysis to ordinary multivariable calculus. To find u and v, just compute f(x + i y) and separate the real and imaginary pieces. Take f(z) = z^2: then f(x + i y) = (x + i y)^2 = x^2 - y^2 + i (2 x y), so u(x, y) = x^2 - y^2 and v(x, y) = 2 x y. The pair (u, v) is everything f does, written in real coordinates.
This bookkeeping is the bridge to nearly every deeper idea. Continuity of f will turn out to mean continuity of both u and v; the Cauchy-Riemann equations (u_x = v_y and u_y = -v_x) are conditions linking these two parts; and harmonic functions appear because, for a nice f, both u and v satisfy Laplace's equation u_xx + u_yy = 0. Learning to pass fluently between 'f' and 'the pair (u, v)' is a core skill.
For f(z) = z-bar (the conjugate), f(x + i y) = x - i y, so u(x, y) = x and v(x, y) = -y. Plain as they look, this pair will later fail the Cauchy-Riemann test, which is why z-bar is not complex-differentiable.
Separating into (u, v) is the routine first move that connects f to real-variable calculus.
Sometimes it is far cleaner to use polar coordinates and write f = u + i v with u and v depending on r and theta instead of x and y. The split is the same idea; only the coordinates change.