I think Geometric Algebra [1] provide a more natural approach to "imaginary" numbers than Gauss does above.
Not only does these algebras give a more intutive understanding of "imaginary" numbers as rotation in a plane (and not simply an alternative R2).
They also extend nicely into all sorts of applications in Physics, Machine learning, etc where Lie groups are needed.
And there is nothing preventing us from defining e1*e2 as i, and use the regular notation for Complex Analysis where the Group Theory aspects are not needed.
Not only does these algebras give a more intutive understanding of "imaginary" numbers as rotation in a plane (and not simply an alternative R2).
They also extend nicely into all sorts of applications in Physics, Machine learning, etc where Lie groups are needed.
And there is nothing preventing us from defining e1*e2 as i, and use the regular notation for Complex Analysis where the Group Theory aspects are not needed.
[1] https://www.youtube.com/watch?v=PNlgMPzj-7Q