TITLE: Can Euler's identity be extended to quaternions?
QUESTION [42 upvotes]: Euler's identity is $e^{i \pi} + 1 = 0$, a special case of the Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$, where $\theta = \pi$ (half-turn of the unit circle).
It is commonly described as "beautiful" because it is simple and includes a bunch of fundamental mathematical constants and operators, including the complex numbers.  
Is there a corresponding Euler's identity for quaternions?
Speculation: 
The Tau Manifesto proposes a different fundamental circle constant, $\tau = 2 \pi$, and a corresponding alternate Euler identity, based on a complete turn, the special case of $\theta = \tau$: 
$e^{i\tau} = 1$
If both identities are extended to quaternions, is one any simpler or more "beautiful" than the other?

REPLY [4 votes]: (It is rather a comment and directly follows from previous answers) 
We could say briefly using Hamilton terminology: 
For any “unit right quaternion” $r$ (i.e., $r=xi +yj+zk$, $x^2+y^2+z^2=1$) it is true an analogue of the Euler's identity $e^{r\pi}+1 = 0$. 
Right quaternions now also called “pure imaginary”.