What does $E^{\mathbb{C}}$ mean?

Question

I was reading a book (Symplectic Geometry and Quantum Mechanics) and find it hard to understand this following example:

Definition: a "complex structure" on a vector space $E$ is any linear isomorphism $j:E \rightarrow E $ such that $j^2 = - I$.

Example: Let $j$ be a complex structure on a vector space $E$. Show that one can define on $E$ a structure of complex vector space $E^{\mathbb{C}}$ by setting $$(\alpha + i \beta)z = a + \beta j z.$$

I don't quite understand what is the meaning of the symbol $E^{\mathbb{C}}$ by putting $\mathbb{C}$ on the right top position. What is the meaning of a symbol when a set is put on the right top position?

Answer 1

Edit: Okay, now I think it just means "$E$, but regarded as a complex vector space."