Is it correct to say the field of complex numbers is contained in the field of quaternions?

The field of complex numbers is isomorphic to a subfield of the division ring of the quaternions. Also, the field of real numbers is isomorphic to a subfield of the complex field.

Whether or not we have subsets here depends on the way the complex numbers and the quaternions are defined.

The ring of quaternions(it is not a field) contains infinitely many subrings that are fields and are isomorphic to the field of complex numbers.

The quaternion skew field ${\Bbb H} = {\Bbb C}\times {\Bbb C}$ has ${\Bbb C}$ as a subfield by considering the ring monomorphism ${\Bbb C}\rightarrow {\Bbb H}: z\mapsto (z,0)$.

Here the addition is defined component-wise and the multiplication is defined as $$(z,t)(z',t') := (zz' - t^* t' , z^*t' + tz' ),$$ where $*$ means conjugation.

Thus $(z,0)+(z',0) = (z+z',0)$ and $(z,0)(z',0) = (zz',0)$ as required.