I've seen lecturers refer to "complex dimension" where a 2D complex number has a complex dimension of 1. I think Penrose uses this idea when talking about spinors.
The phrase complex numbers of a higher dimension is a bit of misnomer. When you try to extend fields to higher dimensions it generally doesn't work and even when it does, i.e. quaternions, the field elements are not complex numbers, they are quaternions, and they don't obey all the same laws as complex numbers do. Multiplication for example isn't commutative over the quaternions I believe.
One of the real math experts here can expand on this I'm sure.
In the meantime you can read this.