TITLE: A new $4$-dimensional number system?
QUESTION [20 upvotes]: I want to create a new number system with the following rules:

The number system is 4-dimensional over the reals, with $\{1,i,j,k\}$ as a basis.
$i^2=j^2=k^2=1$
$ij=k$
The number system is a commutative ring.

Is this number system valid? If so, does it have a name? And if not, why not? I tried to look for it online, but the closest I could get was the hyperbolic quaternions (which violate rule 4):

REPLY [5 votes]: If you want your new system to have "nice" properties then the possibilities are severely limited by the Frobenius Theorem.  
Being more precise about "nice", if you want a field so that familiar properties such as division and commutative multiplication are retained then the only finite dimension extension of the the reals is the complex numbers.  
If you want to retain division but are willing to sacrifice commutativity then you have one more finite dimension extension: the quaternions.
Yours is not either of those so it must have some "not nice" property.  As others have said, this is the presence of zero divisors.  In your system, you could not deduce $x = y$ from $ax = ay$ even if you knew that $a \neq 0$.
You can go further but you need to give up more nice properties.  E.g. the Octonians and Sedenions but you have now lost associativity.