Just to elaborate on what is already in the comments, the algebra automorphisms of $\mathbb H$ act transitively on the set of pairs $(u,v)$ where $u$ and $v$ are imaginary quaternions of unit length that are orthogonal to one another.
To see this, I will include here some remarks on $\mathbb H$ and its automorphisms. Part of the OP's concern seems to be that it is not a priori automatic that metric concepts in $\mathbb H$ such as unit length or orthogonality (and hence the notion of being imaginary, since the imaginary quaternions are the orthogonal complement to $\mathbb R$ in $\mathbb H$)
are preserved by Aut$(\mathbb H)$, and so one of my goals is to show that this concern is not necessary. Indeed, this geometry is intrinsic to the quaternions, as we will see.
(This is not coincidence: Hamilton was led to his discovery by trying to algebraize the geometry of $\mathbb R^3$.)
Note first that imaginary quaternions are characterized by the condition that
$\overline{u} = - u$, and thus for such quaternions, $|u|^2 = -u^2$. Thus if $u$ is imaginary, $u^2$ is a non-positive real number. Converesly,
if $u^2$ is a negative real number,
then one sees that $u$ is imaginary (exercise), and so the imaginary quaternions are also
characterized by having non-positive real squares. In particular, the set of imaginary
quaternions is preserved by Aut$(\mathbb H)$.
On imaginary quaternions, the inner product $u\overline{v} + v\overline{u}$ is simply
$u v + u v$, and so is also preserved by Aut$(\mathbb H)$. In particular, metric concepts like "length one" and "orthogonal" are preserved by Aut$(\mathbb H)$.
If $u$ and $v$ are unit length orthogonal imaginary quaternions, we then have that
$u^2 = v^2 = -1$ (unit length condition) and that $u v = - v u$ (orthogonality condition).
Thus, from the defining relations of $\mathbb H$, we obtain an algebra map
$\mathbb H \to \mathbb H$ that maps $i$ to $u$ and $j$ to $v$ (and then $k$ to $u v$).
This map is non-zero (since $u$ and $v$ are non-zero, having unit length), and hence
is necessarily injective ($\mathbb H$ is a division ring, hence has no non-trivial ideals),
and thus in fact bijective (source and target are of the same dimension).
An automorphism of $\mathbb H$ is determined by its values on $i$ and $j$ (since they generate
$\mathbb H$), and so the previous discussion shows that in fact Aut$(\mathbb H)$ is the
same as the group or permutations of pairs $(u,v)$ of orthogonal pairs of unit
vectors in the imaginary quaternions (also known as $\mathbb R^3$).
This group is well-known: it is precisely $SO(3)$. (If you like, $u$ and $v$ determine
uniquely a mutually orthogonal vector --- their quaternionic product $u v$ --- which can be characterized geometrically in terms of $u$ and $v$ via the right hand rule; thus pairs $(u,v)$ are the same as positively oriented orthonormal bases of $\mathbb R^3$, permutations of which are precisely the group $SO(3)$.)
Incidentally, it is not coincidence that Aut$(\mathbb H) = SO(3)$.
Namely, there is a natural map $\mathbb H^{\times} \to $ Aut$(\mathbb H)$ (where
$\mathbb H^{\times}$ means the non-zero --- equivalently invertible --- quaternions),
given by mapping $q$ to the automorphism $x \mapsto q x q^{-1}$.
The kernel of this map is precisely the centre, and so it induces an injection
$\mathbb H^{\times}/\mathbb R^{\times} \hookrightarrow $ Aut$(\mathbb H)$.
Now the source of this map can be identified with the quotient of the unit quaternions
(which form a copy of $SU(2)$) by $\pm 1$, and of course $SU(2)/\{\pm 1\} =
SO(3)$. On the other hand, this injection is in fact a bijection (i.e. any automorphism
of $\mathbb H$ is inner), by the Skolem--Noether theorem. This puts the description of Aut$(\mathbb H)$ obtained above into a more general perspective.
If $F/{\mathbf Q}$ is a quadratic field then all but finitely many places $v$ of ${\mathbf Q}$ are unramified in $F$, and we could interpret what that means in a couple of ways: prime ideal factorization (for nonarchimedean $v$), extensions of absolute values (any $v$), or base extension by ${\mathbf Q}_v$ (any $v$). Let's use the last way: we look at ${\mathbf Q}_v \otimes_{\mathbf Q} F$. First suppose $v$ is nonarchimedean. For unramified $v$, the tensor product is ${\mathbf Q}_v \times {\mathbf Q}_v$ (if $v$ splits) or it is an unramified quadratic extension of ${\mathbf Q}_v$ (if $v$ is inert). For ramified $v$, the tensor product is a ramified quadratic extension of ${\mathbf Q}_v$ in the sense of ramified extensions of complete discretely valued fields. Next suppose $v$ is the archimedean place of ${\mathbf Q}$. We call $v$ unramified in $F$ when ${\mathbf Q}_v \otimes F$ is ${\mathbf R} \times {\mathbf R}$ ($F$ is real quadratic), and call it ramified in $F$ when ${\mathbf Q}_v \otimes F$ is $\mathbf C$ ($F$ is imaginary quadratic).
When $B$ is a quaternion algebra over ${\mathbf Q}$ and $v$ is a place of ${\mathbf Q}$, the base extension ${\mathbf Q}_v \otimes_{\mathbf Q} B$ is the matrix algebra ${\rm M}_2({\mathbf Q}_v)$ except for finitely many $v$, when it is a division algebra (something subtle). Think of ${\rm M}_2({\mathbf Q}_v)$ as being a noncommutative analogue of ${\mathbf Q}_v \times {\mathbf Q}_v$, and this is a reason for calling ${\rm M}_2({\mathbf Q}_v)$ the unramified case while the division algebra case is called ramified since those are the finitely many peculiar cases. The situation with quaternion algebras doesn't have anything like the inert case from quadratic fields, so "split" and "unramified" are used as synonyms (we say $v$ is "split" or "unramified" in $B$ when ${\mathbf Q}_v \otimes_{\mathbf Q} B$ is a matrix algebra, and "nonsplit" or "ramified" otherwise).
An analogy with quadratic extensions $L$ of $K :={\mathbf C}(X)$, where all residue fields are algebraically closed, is useful to bear in mind. For a discrete valuation $v$ on $K$ that is trivial on the constants, there is a notion of ramified or unramified for $v$ in $L$. It could be defined in terms of how the prime ideal associated to $v$ in the localization ${\mathcal O}_v$ of $K$ decomposes in the integral closure of ${\mathcal O}_v$ in $L$. Or it could be defined in terms of the base extension of $L$ by the completion of $K$ at $v$,$K_v \otimes_{K} L$: if this is $K_v \times K_v$ then we call $v$ unramified or split in $L$ (the two words are synonyms here) and otherwise this base extension is a quadratic ramified extension of $K_v$ (in the sense of ramified extension of a complete discretely valued field) and we then call $v$ ramified or nonsplit in $L$.
Best Answer
In the early 1900s, Dickson introduced what he called generalized quaternion algebras over any field $K$ of characteristic not 2. These are exactly what we'd call quaternion algebras over $K$. His definition was in terms of a basis with rules for their products, and he gave a criterion for these to be division rings. In particular, these were the first noncommutative division rings besides the quaternions of Hamilton, aside from subrings of Hamilton's quaternions.
Three of Dickson's works where he introduces these algebras are
(1) Linear Algebras, Trans. AMS ${\bf 13}$ (1912) 59-73.
(2) Linear Associative Algebras and Abelian Equations, Trans. AMS ${\bf 15}$ (1914), 31-46.
(3) Algebras and Their Arithmetics, Univ. of Chicago Press, 1923.
In (1) he gives the defining equations for a generalized quaternion algebra and the norm criterion for it to be a division algebra (pp. 65-66), though without using the label "generalized quaternion algebra." He writes in a footnote that this work goes back to 1906.
In (2) he constructs cyclic algebras, without using that name, and calls the special case of dimension 4 a generalized quaternion algebra. (Archaic terminology alert: Dickson refers to equations defining cyclic Galois extensions as uniserial abelian equations.)
In (3) he defines cyclic algebras again without using that name (p. 65), writes them as $D$, refers to them as algebras of type $D$ (p. 68), and remarks in a footnote on p. 66 that Wedderburn calls them Dickson algebras. Near the end of the book he looks at generalized quaternion algebras over the rationals with $\mathbf Q(i)$ as a maximal subfield, and as a particular example he uses the Hamilton quaternions over $\mathbf Q$ to describe all rational and integral solutions of certain quadratic Diophantine equations in several variables. Think about how a sum of four squares factors over the quaternions to imagine how arithmetic properties of quaternions could be useful to analyze a Diophantine equation involving a sum of four squares; this is similar in spirit to the way the Gaussian integers are useful in studying a Diophantine equation involving a sum of two squares. The term "arithmetics" in the title of the book is, as far as I can tell, Dickson's label for what we'd call maximal orders, so the book would be called today "Algebras and Their Maximal Orders."
The motivation for Dickson's interest in quaternion algebras was the earlier development of integral Hamilton quaternions, due first to Lipschitz (all integral coefficients, which is clunky in the same way that $\mathbf Z[\sqrt{-3}]$ is compared to $\mathbf Z[(1+\sqrt{-3})/2]$) and then to Hurwitz (all integral or all half-integral coefficients). To study quadratic Diophantine equations in several variables going beyond a sum of four squares, Dickson was led to extend the original definition of quaternions. His main interest was developing the right theory of generalized integral quaternions, rather than just a theory over a field. A further reference in this direction is Dickson's paper
(4) On the theory of numbers and generalized quaternions, Amer. J. Math ${\bf 46}$ (1924), 1-16.
Concerning the connection to central simple algebras, Dickson was the first to show any division algebra that is 4-dimensional over its center is cyclic, at least outside characteristic 2 since he didn't have a good definition in characteristic 2. Taking into account Wedderburn's theorem that every finite-dimensional central simple algebra over a field is a matrix algebra over a division algebra, and that matrix algebras are cyclic, Dickson had shown that every 4-dimensional CSA over a field not having characteristic 2 is an algebra of "his" type. (Wedderburn proved the analogous theorem for dimension 9.)