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Meet i, write numbers as a + ib, and add, subtract, multiply and divide them. Find powers of i, and the identities that still hold for complex numbers.
Picture complex numbers as points, find their modulus and conjugate and what they mean on the plane, then solve quadratic equations with a negative discriminant and find square roots of complex numbers.
Beyond the textbook: the polar form, multiplication as a rotation, the cube roots of unity, and loci on the Argand plane, the parts that JEE, A-Level Further Maths and IB Higher Level ask about most, with a full timed paper.