The complex field is formed by considering the set of ordered pairs of real numbers $\{(a,b):\,a \in \mathbb{R},\,b \in \mathbb{R}\}$ and assigning upon it the two operations $+$ and $\cdot$ defined as follows $$(a,b)+(c,d)=(a+c\,,\,b+d)$$ $$(a,b)\cdot(c,d)=(ac-bd\,,\,ad+bc)$$ My question is, does there exist any useful system where one can generalize this field to obtain a general field $\mathbb{F}$ by considering the set of ordered n-tuples of real numbers $\{(a_1,\ldots,a_n):\,a_i \in \mathbb{R},\,i=1,\ldots,n\}$ and assigning upon them some properly defined operations $+$ and $\cdot$ such that $(\mathbb{F},+,\cdot)$ is a field?
My guess is that, either any system with such a field should simplify down to a system which is based on the usual real or complex fields, or such a generalized field will be too weird to have useful applications. But I cannot find a precise argument for that. Any clarification in this regard will be much appreciated. Thanks in advance.
Frobenius proved that
The more general notion is hypercomplex number, which sacrifices other properties such as associativity.