Background
I was fiddling around and wondered about the below.
Consider the following quaternions:
$$i^2 = j^2 = k^2 = -1$$
Consider the analog of the line element:
$$ ds = dx i + dy j + dz k + \alpha dt$$
where $\alpha$ is a gauge function. Squaring both sides:
$$ ds^2 = - dx^2 - dy^2 - dz^2 + \alpha^2 dt^2 + 2 \alpha dx dt i + 2 \alpha dy dt j+ 2 \alpha dz dt k$$
We choose $\alpha$ such that:
$$ \alpha^2 + 2\alpha \frac{dx}{dt} i+ 2\alpha \frac{dy}{dt} j+ 2\alpha \frac{dz}{dt}k = c^2$$
Hence,
$$ ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 $$
Question
Is it possible to formulate general relativity this way? Have people already combined general relativity and quaternions?