They are separate concepts, it's just that hypercomplex numbers can be used to model the behavior of vectors in $\mathbb{R}^n$.
A vector is an element of a vector space. Vector spaces are (up to isomorphism) direct sums of the field $k$, in that a vector is an ordered $n$-tuple $(a_1,\ldots,a_n)$ with each $a_i \in k$ which can be multiplied by an element $b \in k$, which is called a scalar, as follows: $k*(a_1,\ldots,a_n) = (k*a_1,\ldots,k*a_n)$. In the case of real vectors, this field is $\mathbb{R}$.
Hypercomplex numbers, such as the quaternions, are different in that they are elements of certain algebras over the real numbers. Algebras are defined similarly to vector spaces, only they also allow you to multiply elements of the algebra by other elements of the algebra.
Essentially, vectors and hypercomplex numbers are different concepts and cannot be used interchangeably. However, they are similar in that often the same problem can be solved using either.