You say that i is "the square root of -1", but which one is it? There are two. This is the point in the essay---we cannot tell the difference between i and -i unless we have already agreed on a choice of which square root of -1 we are going to call i. Only then does the other one become -i. How do we know that my i is the same as your i rather than your -i?
To fix the coordinate structure of the complex numbers (a,b) is in effect to have made a choice of a particular i, and this is one of the perspectives discussed in the essay. But it is not the only perspective, since with that perspective complex conjugation should not count as an automorphism, as it doesn't respect the choice of i.
Is it two, or is it infinite? The quaternions have three imaginary units, i, j, and k. They're distinct, and yet each of them could be used for the complex numbers and they'd work the same way. How would I know that "my" imaginary unit i is the same as some other person's i? Maybe theirs is j, or k, or something else entirely.
One perspective of the complex numbers is that they are the even subalgebra of the 2D geometric algebra. The "i" is the pseudoscalar of that 2D GA, which is an oriented area.
If you flip the plane and look at it from the bottom, then any formula written using GA operations is identical, but because you're seeing the oriented area of the pseudoscalar from behind, its as if it gains a minus sign in front.
This is equivalent to using a right-handed versus left-handed coordinate systems in 3D. The "rules of physics" remain the same either way, the labels we assign to the coordinate systems are just a convention.
There are 2 square roots of 9, they are 3 and -3. Likewise there are two square roots of -1 which are i and -i. How are people trying to argue that there are two different things called i? We don't ask which 3 right? My argument is that there is only 1 value of i, and the distinction between -i and i is the same as (-1)i and (1)i, which is the same as -3 vs 3. There is only one i. If there are in fact two i's then there are 4 square roots of -1.
Notably, the real numbers are not symmetrical in this way: there are two square roots of 1, but one of them is equal to it and the other is not. (positive) 1 is special because it's the multiplicative identity, whereas i (and -i) have no distinguishing features: it doesn't matter which one you call i and which one you call -i: if you define j = -i, you'll find that anything you can say about i can also be shown to be true about j. That doesn't mean they're equal, just that they don't have any mathematical properties that let you say which one is which.
Your view of the complex numbers is the rigid one. Now suppose you are given a set with two binary operations defined in such a way that the operations behave well with each other. That is you have a ring. Suppose that by some process you are able to conclude that your ring is algebraically equivalent to the complex numbers. How do you know which of your elements in your ring is “i”? There will be two elements that behave like “i” in all algebraic aspects. So you can’t say that this one is “i” and this one is “-i” in a non arbitrary fashion.