As we know Q IS CONTAINED IN R AND R IS CONTAINED IN C (AS A FIELD)
THERE ARE SOME RESULTS WHICH HOLD IN R BUT NOT IN C ( For example: Order property hold in R but not in C). ALSO THERE ARE RESULTS WHICH HOLD IN C BUT NOT IN R ( For example: Fundamental Theorem of Algebra hold in C but not in R).
For example, a polynomial of order n always has n roots in C, but not in R. Some roots may be complex. Example: x^2 +1=0.
Both Q and C have completions that have non trivial continuous automorphisms. For Q these are the p-adic numbers Qp which have a Frobenius automorphism. For C the only completion is C itself and the automorphism is complex conjugation. On the other hand, the only completion of R is R itself and it has no non trivial automorphism.
This is a good question with lots of interesting twists and turns.
For example, see THEOREM 4.7.15. There exist some isometric embeddings `
\[
\mathscr{l}^m_{2;K} → \mathscr{l}^n_{p;K} corresponding to the triples (m, p, n)
\]
on page 178 and THEOREM 4.7.20. on page 179, for K=R, there are many values of m,p,n that do not hold for the K=C case in
Oksana Shatalov, ISOMETRIC EMBEDDINGS $\mathscr{l}^m_{2} → \mathscr{l}^n_{p} $ AND CUBATURE FORMULAS OVER CLASSICAL FIELDS, Ph.D. thesis, TECHNION:
http://www.math.tamu.edu/~shatalov/MAINPH.pdf
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