The arbitrary of assigning the labels $\mathbf{i}$ and $\mathbf{-i}$ I have heard "Every equation involving complex numbers and expressions retains its truth value if every complex number/variable is replaced by its complex conjugate."
But, I don't understand how this can be the case. An intuitive reason I recently checked up on is that "the labels $\mathbf{i}$ and $\mathbf{-i}$ are really arbitrary, there is no way to know which is which." 
I am not sure I follow. 
Just because 


*

*$-(-i) = i$

*$i, -i$ are roots of the same equation $x^{2} + 1 = 0.$


doesn't prove that $\mathbf{i}$ and $\mathbf{-i}$ are truly arbitrary, right?
So, why are the labels $\mathbf{i}$ and $\mathbf{-i}$ arbitrary? Is there any intuitive way of understanding this?
 A: The difference between the labels $i$ and $-i$ is whether or not you like to draw your $y$-Axis from bottom to top or from top to bottom. It is the difference whether or not you say clockwise or counter-clockwise the mathematical positive direction of rotation. A matter of convention.
A: I think the statement carries less meaning that a first reading suggests.
I would interpret this is the following way: You are given some variables $z_1,...,z_n$ and an equation. Let $A$ be the set of values of $(z_1,...,z_n)$
such that the equation is true.
Then $(z_1,...,z_n) \in A$ iff $(\overline{z_1},...,\overline{z_n}) \in \overline{A}$.
A: Depending on the algebraic structure under consideration, we have units which are elements $u_i$ such that in the same structure is an element $u_i^*$ such that $u_i\cdot u_i^* = 1$. As our algebraic structure gets more involved we may make other demands like the norm of $u_i$ must be $1$. In a sense not unlike how we can factorize integers uniquely into primes up to ordering we can say things like the GCD of 24 and 14 is 2 up to multiplication by a unit since the set of all multiples of $-2$ is the same as the set of all multiples of $2$ (in, say, $\mathbb{Z}$). This doesn't mean all units are indistinguishable. $i$ is different from $-i$ without question.
