are you sure that d:S x S --> C? just saying, the complex field comes with a "natural metric" defined in terms of complex numbers, but which in fact is always real-valued:

d(z,w) = |z - w|. i am suspecting that instead of a metric, what is actually being defined is an inner product, which IS a function from SxS to C.

given an inner product, one can use the inner product to define a norm by |φ| = √(<φ|φ>) (or, in more standard mathematical notation, |φ| = √<φ,φ>).

a metric is then defined in terms of the norm.

Hilbert spaces can be over the reals or the complex field, but the field must be (topologically) complete. taking the field to be the complex field

is "more general" because most of the differences between the two cease to exist when the scalar is taken to be real. for example,

the complex inner product is sesquilinear, but the difference between that and bilinear disappears when the imaginary part is 0.

inner product spaces can only be defined over a field with an ordered subfield. this automatically forces the field to be of characteristic 0.

to define the norm as a square root, the field must be quadratically closed (every number needs to be a square).

Hilbert spaces require that Cauchy sequences of vectors converge, which means that the field has to be complete, and therefore has to contain

a complete ordered field, that is, the real numbers. so you have the demands of the algebra on one hand, and the demands of the topology

on the other. the complex field is particularly well-suited to these demands.

it is possible to have metric vector spaces over other fields.