[jknz]

A field is more delicate if it needs to be enlarged (into a larger field) for all polynomials with coefficients in the field to have roots. There is rich and intricate structure in the possible successive enlargements.

For the field C of complex numbers, all polynomials already have roots so there is no need to enlarge. In that sense the complex field has much less structure than the smaller ones (Z, Q, R).