If you ask for a purely algebraic proof, algebraic K-theory is (almost) OK. Indeed for a compact Hausdorff space $X$, it's topological K-theory $K(X)$ is the same as algebraic K-theory $K_0(C(X;\mathbb{C}))$. So rewrite the proof of Bott-Milnor-Kervaire we will get a (almost) purely algebraic proof.
EDIT(More detail):
The method topologists solve this problem (according to my knowledge) is constructing an complex line bundle $\xi$ over real projective space $\mathbb{R}\mathbb{P}^{n-1}$, and claim that $n(\xi-\varepsilon)=0$ in $K(\mathbb{R}\mathbb{P}^{n-1})$, where $\varepsilon$ is the (complexification of) trivial line bundle over $\mathbb{R}\mathbb{P}^{n-1}$. Since we can compute $K(\mathbb{R}\mathbb{P}^{n-1})=\mathbb{Z}/2^{[\frac{n-1}{2}]}\mathbb{Z}$ with a generator $\xi-\varepsilon$, we must have $2^{[\frac{n-1}{2}]}|n$, and thus $n=1,2,4,8$.
Now let $A=C(\mathbb{R}\mathbb{P}^{n-1};\mathbb{C})$, the ring of continuous functions from $\mathbb{R}\mathbb{P}^{n-1}$ to $\mathbb{C}$. As above we can construct a finite generated projective module $\Gamma(\xi)$ over $A$ and so on. The idea of topological proof now becomes purely algebraic except that the concept of continuous.
Clearly you are assuming some kind of finite-dimensionality over the center.
To classify the finite-dimensional associative division algebras over $\mathbf Q_p$, or more generally over a local field, it's standard to fix the center. A $K$-central algebra here will mean a $K$-algebra whose center is $K$, so $\mathbf C$ and ${\rm M}_2(\mathbf C)$ are each $\mathbf C$-central algebras, not $\mathbf R$-central algebras. And rather than say each finite-dimensional associative $\mathbf R$-division algebra (meaning $\mathbf R$ is in the center but possibly not the whole center) has to be $\mathbf R$, $\mathbf C$, or $\mathbf H$, it would be better to say the only finite-dimensional associative $\mathbf R$-central division algebras are $\mathbf R$ and $\mathbf H$, while the only finite-dimensional associative $\mathbf C$-central division algebra is $\mathbf C$.
Associative division algebras having center equal to a local field and being finite-dimensional over the center are discussed in Pierce's book Associative Algebras. Chapter 15 is on cyclic division algebras and chapter 17 is on division algebras over local fields. Don't expect an easy method to determine which cyclic algebras are division algebras in general, but it's possible to give an easy method "in principle" in special cases. For instance, if $A$ is an $F$-central simple algebra with $\dim_F(A) = p^2$ for a prime number $p$, then either $A \cong {\rm M}_2(F)$ or $A$ is a division ring. In practice you may need to some work to figure out if such a central simple algebra given to you in an abstract form is or is not the matrix ring.
For a field $F$, a quaternion algebra over $F$ is defined to be an $F$-central simple algebra of dimension $4$. An example is ${\rm M}_2(F)$, and sometimes it is the only example ($F = \mathbf C$ and $F$ finite). We call ${\rm M}_2(F)$ the "split" or "trivial" quaternion algebra over $F$. All other quaternion algebras over $F$ are division rings, and when $F$ is $\mathbf Q_p$ or any other local field there is one nontrivial quaternion algebra over $F$. Over $\mathbf Q_2$ this algebra is $\mathbf H(\mathbf Q_2)$, but for $p > 2$ we have $\mathbf H(\mathbf Q_p) \cong {\rm M}_2(\mathbf Q_p)$. A uniform description of the nontrivial quaternion algebra over $\mathbf Q_p$ for all $p$ uses a cyclic algebra construction based on the quadratic unramified extension of $\mathbf Q_p$. (Note: there are infinitely many nonisomorphic quaternion algebras over $\mathbf Q$. The contrast between that and finiteness of the number of quaternion algebras over $\mathbf R$ and $\mathbf Q_p$ is analogous to the contrast with quadratic extension fields: $\mathbf R$ and each $\mathbf Q_p$ have only finitely many quadratic extension fields up to isomorphism, while $\mathbf Q$ has infinitely many.)
You could call the unique nontrivial quaternion algebra over $\mathbf Q_p$ "the $p$-adic quaternions" but that label is not standard. It's more often called the nonsplit or nontrivial quaternion algebra over $\mathbf Q_p$.
The recent book by John Voight on quaternion algebras has an account on quaternion algebras over local fields in Chapter 13.
A brief account on the history of the quaternion algebra construction over fields other than $\mathbf R$ is described here.
Best Answer
A MathSciNet search reveals a paper by Urbanik and Wright (Absolute-valued algebras. Proc. Amer. Math. Soc. 11 (1960), 861–866) where it is proved that an arbitrary real normed algebra (with unit) is in fact a finite-dimensional division algebra, hence is one of the four mentioned in the OP. A key piece of the argument (Theorem 1) is to show that such an algebra $A$ is algebraic, in the sense that if $x \in A$, then the subalgebra of $A$ generated by $x$ is finite-dimensional. The authors then invoke a theorem of A. A. Albert stating that a unital algebraic algebra is a finite-dimensional division algebra.