"Vector" is a fairly abstract algebraic term, that is used for a variety of different purposes. For instance, one example of vectors is the functions who's output is a real number. The point here is that "vector" is far more general than quaternions, so quaternions will not fully replace the need for vectors. Also, quaternions are an example of vectors, so it doesn't really make sense to call them an alternative)

Getting back to your original question, it might make things easier to look at a simpler example. Imagine all we had was the rational numbers, called Q. Then Pythagoras comes along and proves that sqrt(2) is not rational.

To fix this, we introduce the new number √2; and define a new number system where all number take the form a+b√2. To formalize this system, we say that what these numbers are actually 2 dimensional vectors (a,b) with addition and multiplication defined as follows:

(a,b) + (x,y) = (a+x, b+y)
(a,b) * (x,y) = (ax+2by, bx+ay)

We can call this new number system Q(√2), to reflect the fact that we got it by adding √2 to Q.

Note that this is almost the exact vector definition for complex numbers, except that we have a "2" instead of "-1" when we define multiplication.

Now, having done that, suppose Pythagoras comes back and points out that we are still missing the square root of 3. We can use the same trick to introduce √3

(a,b) + (x,y) = (a+x, b+y)
(a,b) * (x,y) = (ax+3by, bx+ay)

Giving us Q(√2)(√3)

In this case, a,b,x,y are not rational numbers, but elements of Q(√2). In this was, elements of Q(√2)(√3) can themselves be seen as 2 dimensional vectors of elements of Q(√2).

On the other hand, if we wanted to, we could also define Q(√2)(√3) directly as a 4 dimensional vector over Q, which we can express as Q(√2,√3) to reflect the fact that we are constructing it in a single step:

(a,b,c,d) + (w,x,y,z) = (a+w, b+x, c+y, d+z)
(a,b,c,d) * (w,x,y,z) = (aw+2bx+6c7+3dz, ax+bw+3cz+3dy, ay+bz+cw+dx, az+2by+2cx+dw)

Assuming I didn't make a mistake typing out the above, you can verify that it does infact represent the number system where all numbers take the form:

a+b√2 + c√2√3 + d√3

And addition and mulitplication behave as you would expect.

The thing to notice here is that, while it is straightforward to work out what the multiplication rule for Q(√2,√3) is, it is far from a simple rule. On the other hand, by instead thinking of it as Q(√2)(√3), we can break it down into 2 relativly simple multiplication rules.

Simmilarly, if we were to start with Q(√2,√3), we could then view Q(√2) not as a 2 dimensional vector over Q, but instead as a portion of Q(√2,√3) where all the elements take the form (a,b,0,0). As a practical matter, the rule for multiplying in Q(√2,√3) is more complicated then the rule for multiplying in Q(√2), so we tend to prefer to stick with the simpler system until we have a need to go to the more complicated system.

In the same way, you can define quaternions directly as a 4 dimensional extension of the real numbers. However, it generally makes life easier to view them as a 2 dimensional extension of the complex numbers.

As an aside, the complex numbers are often defined not as a simple ordered pair of real numbers, but rather a 2x2 matrix of real numbers that take the form:

a b
-b a

This works out to give us multiplication "for free" as matrix multiplication; where our original definition required us to invent a new defintion of multiplication.

The quaternions are similarly defined as complex values matricies of the form:

a+bi c+di
-c+di a-bi

Or, if you are familiar with the complex conjugate

a b
-b‾ a‾

Again, you get multiplication "for free" from matrix multiplication.

In contrast, if you wanted to define the quaternions directly from the real numbers, they would take the form:

a b c d
-b a -d c
-c d a -b
d -c b a

Which, again is more complicated then what you get by going through the complex numbers.