This was a popular pursuit in the 19th century, called hypercomplex numbers, based on the successful example of quaternions and the more esoteric case of octonions. Hypercomplex numbers are now called "finite dimensional associative algebras" but of such algebras, only quaternions and octonions retain features that resemble complex numbers. Matrices form another algebra similar to complex numbers in important ways, and have their own geometric interpretation different from that of complex numbers and quaternions.
Quaternions are related to 3-dimensional rotations and this aspect is generalized to all higher dimensions by Clifford algebras and the representation theory of the $n$-dimensional rotation group.
To answer the EDIT, there are no nontrivial 3-dimensional examples but you can write down trivial examples by listing the multiplication table. Take $1$ and $i$ with the multiplication law the same as in the complex numbers, and add a new element $N$ with some rule for how to multiply it by $1$ and by $i$, such as $N 1 = 1 N = iN = Ni = N$. This gives an associative commutative multiplication law on "3-d numbers" of the form $a.1 + b.i + c.N$, which can be thought of as a rule for combining triples $(a,b,c)$ and $(a',b',c')$ into a third triple. Examples like this are easy to write down, but there is no rule of combination for triples that goes beyond a minor variations on real or complex numbers. Another example of a trivial variation is to multiply the first two components as complex numbers and the third as real numbers. This is called the "direct product" or "direct sum" of the real and complex number system but there is no new structure there.
In dimension 4 there appear new and interesting examples, the quaternions and the algebra of 2x2 matrices.