If you change the field you change the vector space.
The vector space is defined by the triad: field + set + operations, if you change one of these (in your case field), you change the space, so you can not have an endormorphism.
I am in this hypothetical vector space over A complex field which there is a passage or 'transition' from this complex field into a real field
Complex Field $\rightarrow$ Real Field
I'm not interested to change the vector space but seems that is not possible if you change field because you change vector space if you change field.
Is there any invariant that is preserved when you change the underlying field of a vector space ?
What should be preserved, what operation or structure emerges if we pass from the vector space over a complex field to a vector space over real field or viceversa?