I cannot agree with calling complex numbers just two dimensional.
Function of a complex variable is very different from a function of two variables. You can say these are two different departments of mathematics.
Real numbers are not algebraically complete, but extending it with 'i' makes it complete. Adding another dimension to go to 'two dimensions' does not do anything like this.
Mathematicians are fascinated with complex numbers because it is THE extension of real numbers that completes them in very important sense but it comes with so many unexpected and fascinating properties.
Quantum phase is not two-dimensional, it is complex and it amazes me much much more than two-dimensionality would.
Function of a complex variable is very different from a function of two variables. You can say these are two different departments of mathematics.
Real numbers are not algebraically complete, but extending it with 'i' makes it complete. Adding another dimension to go to 'two dimensions' does not do anything like this.
Mathematicians are fascinated with complex numbers because it is THE extension of real numbers that completes them in very important sense but it comes with so many unexpected and fascinating properties.
Quantum phase is not two-dimensional, it is complex and it amazes me much much more than two-dimensionality would.