the complex plane
Because a complex number z = x + i y carries two independent real numbers, you cannot draw it on a single line — you need a whole flat sheet. The complex plane is exactly that: take the usual xy-plane, call the horizontal axis the real axis (where the ordinary real numbers live) and the vertical axis the imaginary axis (where the multiples of i live), and then z = x + i y is simply the point sitting x to the right and y up. This picture is so useful it has three names: the Argand diagram, the Gaussian plane, or just the complex plane.
Drawing numbers as points unlocks geometry. You can also draw z as an arrow (a vector) from the origin out to the point (x, y); then addition of complex numbers is head-to-tail vector addition, and the length of the arrow is the modulus |z| while the angle it makes with the positive real axis is the argument arg z. Suddenly algebraic facts become visible shapes: the conjugate is a reflection across the real axis, multiplying by i is a quarter-turn counterclockwise, and multiplying by a positive real number stretches the arrow.
This fusion of algebra and geometry is the beating heart of complex analysis. Equations in z describe curves and regions you can see (a circle, a half-plane, a strip), and functions become mappings that move and reshape the plane. Almost every theorem to come has a geometric meaning that the complex plane lets you picture. Keeping a mental picture of the plane will carry you a long way.
The equation |z| = 1 is not a single point — it is the set of all z exactly distance 1 from the origin, i.e. the unit circle. The complex plane lets you 'see' an equation in z as a shape.
Algebraic conditions on z become visible curves and regions in the plane.
The complex plane looks just like R^2, the plane of pairs (x, y), and as a set it is. But the complex numbers carry an extra structure R^2 lacks — a multiplication — and that single extra operation is what makes complex analysis utterly different from plain two-variable calculus.