Is the standard definition of $i$ enough to uniquely describe it? Ahlfors states "From elementary algebra the reader is acquainted with the imaginary unit $i$ with the property $i^2 = -1$." (Complex Analysis, Lars Ahlfors, page 1)
Kreyszig (Advanced Engineering Mathematics) first defines complex numbers “as an ordered pair $(x,y)$”, where $x, y \in \mathbb{R}$. He then defines the imaginary unit $i$ as $(0,1)$. Following this, he defines the addition and multiplication of complex numbers, and uses the definition of multiplication to arrive at $i^2 = -1$. (pages 652, 653)
Professor Paul Dawkins's online notes use both of the above approaches.
Various other places like MathWorld use similar definitions. 
But these definitions only narrow down $i$ to two possible values (which are opposite in sign: $z_1$ and $-z_1$). So, when $i$ is used in an expression or in the representation of a complex number, how does one determine which one of the two values it actually is? Also, the sign of the imaginary part of the complex number will change if one person is using one of the two values given by the definition of $i$ and someone else is using the other one. This ambiguity will also affect the uniqueness of the complex plane.
Keeping the above points in mind, are there additional qualifications that help in narrowing it down to one value? 
Or is it impossible to further narrow it down, and so we say that - we are using one of the square roots of -1 and representing that root as $i$, and we will all use $i$ to mean this same root, although we can't really define which one it is?
 A: The map $$f:\mathbb{C}\rightarrow\mathbb{C}: a+bi\mapsto a-bi$$ (conjugation) is an automorphism of $\mathbb{C}$: if $P$ is a true statement about complex numbers using $+$ and $\times$, then "$f(P)$" is also true, where $f(P)$ is the statement gotten by replacing $z$ with $f(z)$ for every complex number $z$ in $P$. E.g. "$i^3=-1$" is true, and $f($"$i^3=-1$") = "$(-1)^3=-1$" is also true.
So using addition and multiplication alone there is no way to distinguish $i$ from $-i$. That said, there are other operations on $\mathbb{C}$ which "add structure" - e.g. the "imaginary part" function $\mathfrak{Im}: a+bi\mapsto b$ (since we have $\mathfrak{Im}(i)=1$ but $\mathfrak{Im}(-i)=-1$).

Making this a bit more palatable (hopefully!):
We shouldn't really talk about $\mathbb{C}$ as if it were unique. Rather, we should talk about "algebraic closures of $\mathbb{R}$" (which is itself a bit wrong - reall really, we should talk about "algebraic closures of Dedekind-complete ordered fields"). The point is that any two such objects are isomorphic, and that means that when you're holding your version of $\mathbb{C}$ and I'm holding my version of $\mathbb{C}$ we'll never disagree about how they behave - even if they're quite different, or if they're the same except with $i$ and $-i$ "flipped."
The ordered-pairs-of-reals approach describes one way of producing a "version of $\mathbb{C}$."

Ultimately, thinking along these lines will take us to model theory (an area of logic that looks at the problem of what objects/functions/relations/etc. are "definable" in given contexts) and category theory (which can be thought of as studying "purely structural" behavior).
