A very helpful intuition is to consider a notion of "Compass numbers"; one might say, "the nearest coffee shop is East 2 blocks and North 3 blocks". One might

A very helpful intuition is to consider a notion of "Compass numbers"; one might say, "the nearest coffee shop is East 2 blocks and North 3 blocks". One might say the the real numbers are restricted to just two directions. Extend them to allow the "sign" of a number to be any direction. It is intuitive to imagine what addition of "Compass numbers" might look like, and what multiplication by a positive scalar real number might look like. The question is, what should multiplication of two compass numbers neither of which is real be? One of the earliest developments of complex numbers was done by a Danish cartographer, Caspar Wessel. He wrote a lovely paper that was published in an obscure forum in 1797. He is now credited as the first person to understand the correspondence of complex numbers and vectors on a plane. It seems quite natural that a cartographer would be interested in "numbers" that can point in any direction. If you posit the existence of a multiplicative identity and (arbitrarily) label it "1", and then take a compass number 90 degrees away from it and (arbitrarily) label it "i", (and furthermore assume field axioms), the formula "(1 + i) * (1 - i)" forces the conclusion that i * i is the compass number pointing in the opposite direction of the multiplicative identity.