Given that you have a Ph.D. in mathematics, this might seem hopelessly elementary, but who knows--I found it intuitive and insightful: https://betterexplained.com/articles/a-visual-intuitive-guid...
Given that you have a Ph.D. in mathematics, this might seem hopelessly elementary, but who knows--I found it intuitive and insightful: https://betterexplained.com/articles/a-visual-intuitive-guid...
I've always been satisfied with the explanation "Just as you need signed numbers for translation, you need complex numbers to express rotation." Nobody asks if negative numbers are really a natural thing, so it doesn't make sense to ask if complex ones are, IMO.