Gelfand-Mazur complex part mean what it says?

Question

This question pertains to the complex Gelfand/Mazur theorem. Mazur/Gelfand says normed division algebra over the complex numbers is isomorphic to the complex numbers. Therefore, is the statement true: The ONLY normed division algebra over complex numbers is the complex numbers? And, therefore, is this statement true: a normed division algebra over complex numbers is not isomorphic to the reals or quaternions. (There is another post here that appears to confirm the quaternion side of this question but not for the real side.) Some argue that this was not what the theorem meant because the complex numbers can be seen to contain the reals. But that would seem to require the theorem to say that a normed division algebra over the complex numbers is isomorphic to the reals and complex or some kind of transitivity group argument would then put it isomorphic with quaternions (which does not seem to be right.) Clarification: this question is NOT about normed division algebra over real numbers but about normed division algebra over complex.

Answer 1

For Banach Algebra, Wikipedia states, "The only complex Banach algebra which is a division algebra is the complexes. (This is known as the Gelfand–Mazur theorem.)" The "only" was the qualifier this question was asking to be confirmed. (Apologies for asking such a fundamental question.) Also, "Unital Banach algebras over the complex field provide a general setting to develop spectral theory." Rickart, C. E., An elementary proof of a fundamental theorem in the theory of Banach algebras, says, "Mazur-Gelfand ... is directly equivalent to the existence of a spectrum for elements of a normed algebra." Therefore, $\mathbb{R}$ and $\mathbb{H}$ are not complex Banach algebras. If given a Banach algebra over the complex then it is only isomorphic to $\mathbb{C}$.