The exponential extension of $\mathbb{Q}$ is a proper subset of $\mathbb{C}$?

This question come from a recent post Exponential extension of $\mathbb{Q}$.

An exponential field is a field $\mathbb{K}$ where it's well defined a function $E:\mathbb{K} \rightarrow \mathbb{K}$ such that: $$ E(x+y)=E(x)E(y) \quad \forall x,y \in \mathbb{K} \quad \land \quad E(0)=1 $$ In the preceding post, thanks to the answer of Martìn-Blas Pérez Pinilla, it's showed that $ \forall a\in \mathbb{Q}$ we can build an exponential field $\mathbb{E}_a$ in which is well defined the exponential function $E_a(r): \mathbb{E}_a \rightarrow \mathbb{E}_a \;;\; E_a(r)=a^r$. Now consider the field: $$ \mathbb{E}=\left \langle \bigcup_{a \in\mathbb{Q}} \mathbb{E}_a \right\rangle \subset \mathbb{C} $$ that is the field closure in $\mathbb{C}$ of all $\mathbb{E}_a$. This is an exponential field and the question is: is $\mathbb{E}$ a proper subset of $ \mathbb{C}$ or it's $\mathbb{E}= \mathbb{C}$?

Others sub-questions are:

there is a method to determine if a transcendental number $ \in \mathbb{E}$?

the Napier's constant $e$ is an element of $\mathbb{E}$?