The complex numbers are nice in two ways. They are an 'algebraically closed' field, meaning that every polynomial equation with complex coefficients has a complex solution. And they are 'Cauchy complete' metric space, meaning that every Cauchy sequence converges.
We can get the complex numbers in two ways. We can start with the rational numbers and take their Cauchy completion. This gives us the real numbers. But these are not algebraically closed. So we can take their algebraic closure. The result is the complex numbers, which is still Cauchy complete.
Or, we can start with the rational numbers and take their algebraic closure. This gives us the 'algebraic numbers'. There's a way to define a nice metric on these, but the resulting metric space is not Cauchy complete. To fix that, we can take its Cauchy completion. The result is the complex numbers, which is still algebraically closed.
In the first route I used the usual metric on rational numbers. But what if we use one of the p-adic metrics?
We can start with the rational numbers and take their Cauchy completion using the p-adic metric. This gives us the p-adic numbers. But these are not algebraically closed. So then we can take their algebraic closure. There's a nice metric on it, but it's 𝑛𝑜𝑡 still Cauchy complete.
So we can take the Cauchy completion 𝑎𝑔𝑎𝑖𝑛. You may feel sort of pessimistic right around now... but this time the resulting field 𝑖𝑠 algebraically closed, and of course Cauchy complete by definition. So yay, we're done! 🎉
The weird part: the resulting field is isomorphic to the complex numbers equipped with a weird metric. Using the axiom of choice. 😬
