Algebraic structure built from the set of complex numbers Is it correct to say that $\mathbb{C}$ is both a field, and a dimension 2 vector space? The complex numbers can be the elements of the field, or they can be considered as 2 elements of $\mathbb{R}$, in which case $\mathbb{C}$ exhibits the structure of a vector space. 
To generalize, what algebraic structures can we attribute to $\mathbb{C}^{n}$?
So that we are all on the same page (if it was not clear), let's adopt the following standard definitions for field and vector space (there shouldn't be any surprise here, IMHO):
field: a set F together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold: closure of F under + and . ; associativity of + and . ; commutativity of + and . ; existence of identity elements for + and . ; existence of inverses for + and . ; distributivity of . over +. 
vector space: A vector space over a field F is a set V together with two binary operations (+ from VxV to V, * from FxV to F) that satisfy the axioms: associativity, commutativity, existence of identity and inverses for + ; a(u+v) = a u + b v ; (a+b)u = a u + b v ; a(b v) = (ab) v ; 1 v = v ; where a,b,c are in F and u, v are in V.

Frank - you are correct.  $\mathbb{C}$ is a 2-dimensional vector space over $\mathbb{R}$.  Similarly $\mathbb{C}^n$ can be considered an $n$-dimensional vector space over $\mathbb{C}$ and also as a $2n$-dimensional vector space over $\mathbb{R}$.
