Is it closed under addition? yes
Is it closed under scalar multiplication from the field C? no
Not a subfield.
well vector addition is just done componentwise, so if you add two real valued vectors, their sum is clearly still real valued since the real numbers are closed under addition. I mean (a,b,c)+(d,e,f)=(a+d,b+e,c+f) where everything is necessarily real numbers, so its closed under addition. But to be a subspace it also needs to be closed under scalar multiplication, and it is not, this is enough to prove it is not a subspace.
I think it depends on the field.
For example: Every field is a one dimentional vector space over itself.
This implies that has A basis when the scalers are take as complex numbers.
On the other hand has A basis when the scalers are taken as real numbers. It is a two dimentional vectors space over
Now we could view as a closed subspace of the 2nd, but not the first.