The complex numbers are the closure regardless of dimension. When I was writing that I was thinking of the Quaternions, which are the 4 dimensional analog of the complex numbers, in 2^N dimensions this is the Cayley Dickson construction.
The fluke is this: Euclidean space of dimension N has N(N-1)/2 rotational dimensions. If you plug 2 into that you get 2x1/2 which is 1 dimension. i.e. the rotations in 2D space look like a circle. If you add an extra dimension (the radius) you get the polar form of complex numbers.
In other dimensions this doesn't always work. In 3 dimensions we have 3x2/2 = 3 rotational dimensions, so we need a space with dimension 4 (the quaternions). In 4 dimensions we need a 6 dimensional rotation space. We just established that Cayley Dickson algebras only come in powers of 2, so it doesn't fit at all.
But couldn't one use two unit quaternions to describe rotations in 4d space? An antisymmetric matrix in 4 dimensions has N(N-1)/2 = 6 independent variables. But each unit quaternion has three independent variables, so two of them would be enough to describe rotations in 4d.
The fluke is this: Euclidean space of dimension N has N(N-1)/2 rotational dimensions. If you plug 2 into that you get 2x1/2 which is 1 dimension. i.e. the rotations in 2D space look like a circle. If you add an extra dimension (the radius) you get the polar form of complex numbers.
In other dimensions this doesn't always work. In 3 dimensions we have 3x2/2 = 3 rotational dimensions, so we need a space with dimension 4 (the quaternions). In 4 dimensions we need a 6 dimensional rotation space. We just established that Cayley Dickson algebras only come in powers of 2, so it doesn't fit at all.