There have been a number of Scientific discoveries that seemed to be purely scientific curiosities that later turned out to be incredibly useful. Hertz famously commented about the discovery of radio waves: “I do not think that the wireless waves I have discovered will have any practical application.”
Are there examples like this in math as well? What is the most interesting “pure math” discovery that proved to be useful in solving a real-world problem?
What point is there to “prove”? Your argument now is just that we defined them differently therefore they are different. Which suggests a straw man to my original point as I never once implied or suggested that in mathematics, real and complex numbers don’t have different definitions, that’s not relevant to anything.
My point is that the way that you stick two real numbers together to make a complex number is important, and is unintuitive if you approach it as just two real numbers.