What properties do you lose when you extend your number set? [duplicate]

So in $\mathbb{R}$ and $\mathbb{C}$ you have both associative and commutative property, but as you extend to $\mathbb{H}$ you lose the commutative property, and $\mathbb{O}$ loses the associativity. What more can you lose once you extend beyond $\mathbb{O}$? Are there some lesser known or weaker properties which holds for all extensions, or are do all such rules eventually disappear as you extend?

[Also if someone could correctly tag this question, that would be nice]

marked as duplicate by MJD, Venus, Ahaan S. Rungta, Davide Giraudo, NewbJan 9 '15 at 18:42

• I will extend $\Bbb R$ by adding a new element $\star$ with the property that $1 + \star = 2$ and $2 - \star = 17$. What property have we lost? – MJD Jan 9 '15 at 14:52
• @MJD I don't know what this property is called, but $\star$ wouldn't have any inverse in addition. – Frank Vel Jan 9 '15 at 15:06
• You've lost more than that; you've lost the property that if $a+b=c$ then $c-b=a$. Or maybe you've lost the property that every number has an additive inverse. Or maybe you've lost the property of associativity. Or… – MJD Jan 9 '15 at 15:33