This is a follow-up to a previous question regarding complex numbers. Many people there compared complex numbers to vectors, and there was disagreement about what the difference was. Some options people have:
Complex numbers and vectors are exactly identical.
The $i$ in complex numbers changes the way it describes the space, if so, how and why?
Complex numbers are 2D but can be used cleverly to describe larger dimensional systems. If so how and why?
What are the differences?
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The shorthand answer is that complex numbers and $2$-D vectors are different algebraic structures.
Complex numbers are a field in the mathematical sense. That means that there are two operations, addition and multiplication, and they behave like you would expect them to do. You can always multiply two elements and get a new element as an answer.
Vector spaces have addition, but only multiplication by scalar numbers: there is no definition for $\mathbf{v}$ times $\mathbf{w}$ for two vectors, just $a\mathbf{v}$ where $a$ is a scalar (note that scalars are not vectors, as the vector space is defined).
Complex numbers form a field, not a vector space. A basic 2D vector space would be like complex numbers with addition, and multiplication with only real numbers, no general multiplication.
(What if we used complex numbers as scalars in a 2D vector space? Then we get something with 4 dimensions. We could use a 1D vector space with complex scalars, but then we are basically cheating and doing all the work using the scalars.)
In practice we tend to prefer inner product vector spaces, the subset of vector spaces that do have one kind of multiplication of vectors: the dot-product $\mathbf{v}\cdot\mathbf{w}$ that gives us a number. One can also define outer products that give higher order things like matrices, exterior algebras that allow one to make cross products $\mathbf{v}\times\mathbf{w}$ that actually produce vectors. But all of this is extra stuff one needs to decide to add to the basic vector machinery, it does not automatically come with a vector space. And none of it directly corresponds to the vanilla multiplication we want for our complex numbers.
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Complex numbers are not identical to two dimensional vectors, but in some ways complex numbers and vectors have similar behaviours. There are also similarities between complex numbers and two dimensional matrices. Mathematicians call these similarities isomorphisms.
There are two things you can do with a pair of complex numbers. You can add (or subtract) them and you can multiply (or divide) them.
Let's think about addition/subtraction first.
Suppose we map each complex number to a two dimensional vector as follows:
$a+ib \mapsto (a,b)$
Then in the world of complex numbers
$(a+ib) + (c+id) = (a+c) +i(b+d)$
and in the world of vectors
$(a,b) +(c,d) = (a+c,b+d)$
So we can see that
$(a+ib) + (c+id) \mapsto (a,b) + (c,d)$
where the addition on the left hand side is complex number addition, and the addition on the left hand side is vector addition. And we could reverse the mapping and go from vector addition to complex number addition instead.
We say that addition of complex numbers is isomorphic to addition of two dimensional vectors. Note that we are not saying that complex numbers and vectors are the same - we are just saying that for the purposes of addition/subtraction they behave in the same way.
Now let's think about multiplication/division.
When we multiply a complex number by $i$ then we get:
$i(c+id) = -d + ic$
In the vector world we start with a vector $(c,d)$ and end up with a vector $(-d,c)$. We have rotated the vector anticlockwise by $90^o$. We can represent this rotation by the matrix
$\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$
With a little more thought, we can come up with a mapping from complex numbers to two dimensional matrices:
$a+ib \mapsto \begin{pmatrix}a & -b \\ b & a \end{pmatrix}$
and then we have
$(a+ib)(c+id) = (ac-bd) +i(ad+bc)$
and
$\begin{pmatrix}a & -b \\ b & a \end{pmatrix}\begin{pmatrix}c & -d \\ d & c \end{pmatrix} = \begin{pmatrix}ac-bd & -ad-bc \\ ad+bc & ac-bd \end{pmatrix}$
so
$(a+ib)(c+id) \mapsto \begin{pmatrix}a & -b \\ b & a \end{pmatrix}\begin{pmatrix}c & -d \\ d & c \end{pmatrix}$
So we have found another isomorphism. We say that multiplication of complex numbers is isomorphic to multiplication of two dimensional matrices.
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation $x^2 = −1$
Vectors define vector spaces:
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms,
A complicated list of axioms defines vector spaces
From these very general descriptions, complex numbers are a simple extension of real numbers, a new field. Vectors defined in vector spaces, complex numbers and real numbers are a context for the definition of vectors.
Extensions of complex numbers exist, see the answer here, they are just a different field that can be used as the real numbers and complex numbers for multiplying vectors. Read up on the quaternion here .
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One can clearly come up with a 1:1 correspondence between 2D vectors and complex numbers.
Let
$$ \boldsymbol{v} = \begin{bmatrix}x\\y\end{bmatrix} $$
\begin{align} f&:\mathbb{R}^2\rightarrow \mathbb{C}\\ f\left(\boldsymbol{v}\right)&=x+iy \end{align}
This function is injective and surjective so it is a bijection. However, this function doesn't preserve nice properties.
Consider
\begin{align} \boldsymbol{u} = \begin{bmatrix}a\\b\end{bmatrix} \end{align}
\begin{align} f(\boldsymbol{v})\cdot f(\boldsymbol{u}) = x a - yb + i xb + ixa \neq xa+yb = \boldsymbol{v}\cdot\boldsymbol{u} \end{align}
Note on the far left the dot indicates multiplication between two complex numbers and on the far right the dot represents the cross product.
This shows that multiplication between complex numbers is different than dot (or cross) multiplication between vectors.
Just because there is a bijection between two spaces doesn't mean we can think of them as being the same space since they may have different algebraic properties (such as multiplication as shown here) which are not preserved by that bijection.
This isn't to say there aren't important similarities between the two spaces (they are both 2-dimensional, for example) that are worth keeping in mind to help get intuition to solve various problems. However I would say it is going much too far to say that complex numbers and 2D vectors are identical.
Division is not defined for vectors.