You have $\mathbb{R}$ which is a field, i.e. you can add and multiply, and every nonzero elements has a multiplicative inverse.
Now you want to somehow have square roots for all real numbers. So you include (this can be done properly, but it's not the point here) another "number", that we call $i$ and has the property that $i^2=-1$.
Now you still want to have a field; in particular you want to add and multiply, and so you need to make sense of expressions of the form $a+bi$. But it turns out that
$$
\mathbb{C}=\{a+bi:\ a,b\in\mathbb{R}\}
$$
is already a field, i.e. the smallest field that contains both $\mathbb{R}$ and $i$.
The Wikipedia article cites a textbook that manages to confuse the issue further:
Purely imaginary (complex) number : A complex number $z = x + iy$ is called a purely imaginary number iff $x=0$ i.e. $R(z) = 0$.
Imaginary number : A complex number $z = x + iy$ is said to be an imaginary number if and only if $y \ne 0$ i.e., $I(z) \ne 0$.
This is a slightly different usage of the word "imaginary", meaning "non-real": among the complex numbers, those that aren't real we call imaginary, and a further subset of those (with real part $0$) are purely imaginary. Except that by this definition, $0$ is clearly purely imaginary but not imaginary!
Anyway, anybody can write a textbook, so I think that the real test is this: does $0$ have the properties we want a (purely) imaginary number to have?
I can't (and MSE can't) think of any useful properties of purely imaginary complex numbers $z$ apart from the characterization that $|e^{z}| = 1$. But $0$ clearly has this property, so we should consider it purely imaginary.
(On the other hand, $0$ has all of the properties a real number should have, being real; so it makes some amount of sense to also say that it's purely imaginary but not imaginary at the same time.)
Best Answer
We have $a + bi$ is algebraic iff $a$ and $b$ are algebraic.
Therefore, if $a + bi$ is transcendental then at least one of $a$ or $b$ is transcendental.
So, all complex transcendental numbers are "based" on real transcendental numbers.