a complex function
A complex function is just a rule that takes one complex number and gives you back another. You feed in a number z (a point in the plane, written z = x + i y) and the function w = f(z) hands you a new complex number w. It is the same idea as an ordinary function from school — input, rule, output — except that both the input and the output now live in the two-dimensional plane of complex numbers rather than on the one-dimensional number line.
Because the input z sits in a plane and the output w also sits in a plane, you genuinely cannot draw the usual graph: that would need four dimensions (two for z, two for w). Instead you keep two copies of the plane side by side — a z-plane and a w-plane — and think of f as moving points from the first picture into the second. For a concrete feel, take f(z) = z^2. Feed in z = 1 + i; then z^2 = (1 + i)^2 = 1 + 2i + i^2 = 2i, so this f sends the point 1 + i to the point 2i.
This single shift — output is also a plane point, so you watch where points go rather than reading a height off a graph — is the heart of complex analysis. Almost everything later (limits, continuity, the derivative, conformal maps, contour integrals) is best pictured as 'where does f send this point, this curve, this region?' A complex function need not be defined everywhere; the set of inputs it accepts is its domain, treated separately.
Let f(z) = z + 2 - i. Feeding in z = 3 + 4i gives f(z) = 5 + 3i. Every input is simply nudged two units right and one unit down; this rule is a valid complex function even though nothing 'curves'.
Even a plain shift counts as a complex function — the rule just has to assign one output to each input.
A complex function is single-valued by convention: each input z gives exactly one output. Things like the square root or the logarithm assign several values to one input, so they are multivalued and need extra care (a chosen branch) before they count as functions.