Let $x \in \Bbb R^d$ or some other vector space.
Define $\|.\|_c$ as $(\sum_{j=1}^dx_j^c)^{\frac{1}{c}}$ where $c \in \Bbb C$ (Complex number) or $\Bbb H$ (other hypercomplex number systems).
I was playing around with this idea and wondered if anyone knew of any work that has been done on this.
Usually, a modulus of (hyper)complex numbers is defined as follows:
$|z|=\left|\sum_{k=1}^N a_k i_k\right|=\sqrt{\sum_{k=1}^N s_k a_k^2}$
where $s_k$ are the components of the metric signature of the space, and depending on the exact hypercomplex number system, can be either $0$, $1$ or $-1$. This modulus is not, generally, a norm, it is a norm only if all $s_k=1$.
Since this expression can be imaginary (but not negative), and the system in question may not include imaginary numbers (such as split-complex numbers), some authors use the square of the above expression and call it "modulus", so to avoid use of imaginary numbers.