Complex numbers are a two-dimensional field. But other 2d fields can be defined. Edit: That is incorrect, no other 2d scalar field can be defined.
For example, the standard vector addition and the following multiplication would also obey the field definition. $$ \begin{pmatrix}a\\b \end{pmatrix} \cdot \begin{pmatrix}c\\d \end{pmatrix} = \begin{pmatrix}ac\\bd \end{pmatrix}$$
What then makes the complex number multiplication so universally useful? Why are other 2d fields so rarely used and discussed?