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Hurwitz's theorem is stated here (section 2.6 of 'A taste of Jordan algebras' by Kevin McCrimmon) as:

Any composition algebra $C$ over a field $\Phi$ of characteristic not 2 has finite dimension $2^n$ for $n=0,1,2,3$ and is one of the following ...

it goes on then to describe generalisations of the usual normed division algebras.

The wikipedia page composition algebra tells us that one only has a 1-dimensional composition algebra when the characteristic of the base field is not 2, but otherwise you can start from a 2-dimensional composition algebra over a characteristic 2 field and perform the usual Cayley-Dickson construction.

Edit: The following theorem was proved by Kaplansky (Proc AMS 1953), which finishes off the classification. A quadratic form $g$ on an algebra $A$ over a field $F$ in this context is a function $g:A \to F$ such that $g(kx) = k^2g(x)$ for $k\in F$ and $x\in A$.

Theorem. Let $A$ be an algebra with unit element over a field $F$. Suppose that $A$ carries a nonsingular quadratic form $g$ satisfying $g(xy) = g(x)g(y)$ for all $x, y \in A$. Then:
(a) A is alternative,
(b) except for the case where $A$ has characteristic two and is a purely inseparable field over $F$, $A$ is finite dimensional and of dimension 1, 2, 4, or 8,
(c) $A$ is either simple or the direct sum of two copies of $F$,
(d) $g(x) = x*x$ x^\ast x$where$x\mapsto x*$x\ast$ is an involution of $A$.

So unless your base field has characteristic 2, and your division algebra is a purely inseparable extension of the base field, you your division algebra has to be finite dimensional.

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Edit: The following theorem was proved by Kaplansky (Proc AMS 1953), which finishes off the classification. A quadratic form $g$ on an algebra $A$ over a field $F$ in this context is a function $g:A \to F$ such that $g(kx) = k^2g(x)$ for $k\in F$ and $x\in A$.

Theorem. Let $A$ be an algebra with unit element over a field $F$. Suppose that $A$ carries a nonsingular quadratic form $g$ satisfying $g(xy) = g(x)g(y)$ for all $x, y \in A$. Then: (a) A is alternative, (b) except for the case where $A$ has characteristic two and is a purely inseparable field over $F$, $A$ is finite dimensional and of dimension 1, 2, 4, or 8, (c) $A$ is either simple or the direct sum of two copies of $F$, (d) $g(x) = x*x$ where $x\mapsto x*$ is an involution of $A$.

So unless your base field has characteristic 2, and your division algebra is a purely inseparable extension of the base field, you division algebra has to be finite dimensional.

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Hurwitz's theorem is stated here (section 2.6 of 'A taste of Jordan algebras' by Kevin McCrimmon) as:

Any composition algebra $C$ over a field $\Phi$ of characteristic not 2 has finite dimension $2^n$ for $n=0,1,2,3$ and is one of the following ...

it goes on then to describe generalisations of the usual normed division algebras.

The wikipedia page composition algebra tells us that one only has a 1-dimensional composition algebra when the characteristic of the base field is not 2, but otherwise you can start from a 2-dimensional composition algebra over a characteristic 2 field and perform the usual Cayley-Dickson construction.

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