Here are morphisms for various kinds of fields.
Usual arithmetic
Modular arithmetic
Rational function field
Residue class fields
The Gaussian numbers
The embedding of Q in R and of R in C
are morphisms of fields.
Complex conjugation is a morphism c : C -> C.
The subfield {x | c(x) = x} coincides with
R.
Let R be a field, and h 
R[X].
Then the map
R(X) -> R(X),
f(X)/g(X) ->
f(h(X))/g(h(X))
(f, g R[X],
g 
0)
R[X].
Then the map
(f, g
R[X],
g 0)
is a morphism.
Its image is the subfield of R(X) of all fractions
of polynomials that can be written as a polynomial in h.
On the next page, we shall come across examples
of embeddings of one residue class field in another.
As for Q + Qi, we have complex conjugation:
c : Q + Qi -> Q + Qi,
c(a + bi) = a - bi.
The subfield
{x | c(x) = x}
coincides with Q.