Unit complex numbers can be used to represent roatations in 2D, unit quaternions can be used to represent rotations in 3D. Can there be anything ike heptonions which could represent rotations in 4D or it is proven to be impossible? There are 6 degrees of freedom in 4D rotation matrices, so they could be represented well by a sphere in 7-dimensional space.
My attempt:
I tried googling "Heptonions" but I didn't find anything relevant.
The rotation in 4D can be defined by one plane and 2 angles. The plane can be defined by two orthogonal vectors. The first vector can be any unit vector in 4D and the second vector is a unit vector which lies in the space perpendicular to the first vector. However, there are still infinitely many possibilities to define the plane.
No, an arbitrary 4D rotation is defined by a pair of orthogonal 2D plane rotations, so you pick two (oriented) orthogonal 2D planes (this can be done by picking four orthogonal vectors and pairing them up, although of course a single 2D plane can be represented by many pairs of vectors) and an angle for each plane.
That is, every rotation looks like the following in some choice of orthonormal coordinates:
$$ R = \begin{bmatrix} \cos\alpha & -\sin\alpha & 0 & 0 \\ \sin\alpha & \cos\alpha & 0 & 0 \\ 0 & 0 & \cos\beta & -\sin\beta \\ 0 & 0 & \sin\beta & \cos\beta \end{bmatrix}. $$
If $\alpha$ and $\beta$ are distinct the two planes are uniquely determined by the rotation. However if $\alpha=\beta$ (or $\alpha=-\beta$, which means $\alpha=\beta$ if we simply flip one of the planes' orientation) then there are many choices of pairs of orthogonal planes we can use to represent the same rotation $R$.
Quaternions model both 3D and 4D rotations, actually.
Recall a quaternion looks like $a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$, or in other words a combination of a scalar and a 3D vector, and Euler's formula $\exp(\theta\mathbf{u})=\cos(\theta)+\sin(\theta)\mathbf{u}$ holds for unit vectors $\mathbf{u}$ (which are precisely all of the square roots of $-1$ in the quaternions). Set $p=\exp(\theta\mathbf{u})$.
Then $p\mathbf{v}p^{-1}$ is also a vector; it is the rotation of the arbitrary vector $\mathbf{v}$ around the oriented $\mathbf{u}$-axis by the double angle $2\theta$. If $S^3$ denotes the three-sphere of unit quaternions and $\mathrm{SO}(3)$ denotes the group of 3D rotations, there is a $2$-to-$1$ map $S^3\to\mathrm{SO}(3)$. So 3D rotations may be represented by unit quaternions $p$, although $\pm p$ represent the same rotation so there is some redundancy.
For 4D, we can say the space of all quaternions is four-dimensional, and any 4D rotation can be represented by the function $f(x)=axb$ for some unit quaternions $a$ and $b$, again with some redundancy. Topologically, this means the space of all 4D rotations look (almost) like $S^3\times S^3$, not $S^6$ like you suggest.
Octonions don't work as nicely for describing rotations. While multiplying by a unit octonion gives a rotation, these sorts of rotations are not closed under multiplication (that is, the function $a(bx)$ cannot be represented as $cx$ for any $c$, at least in general).
Clifford algebras Angina mentions in the comments generalize even further to represent rotations using algebras that are generated by many anticommuting square roots of $-1$, not just $\mathbf{i}$ and $\mathbf{j}$ like the quaternions are. (Actually they represent "spin groups," whose elements in turn represent rotations, and they can also be used to describe interesting projective representations of rotation groups you can't get from linear representations.)