Misleading wikipedia entry on Banach algebras

The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.

and under counterexamples it states

The algebra of the quaternions $\mathbb {H}$ is not a complex Banach algebra (for any norm on $\mathbb {H}$)

These statements seem contradictory to me. Is that the case?

• According to the answer to this question, the quaternions do form a real Banach algebra but not a complex one. – shalop Aug 17 '17 at 18:59
• Putting the two quoted sentences together, "The quaternions form a 4-dimensional real Banach algebra...", "...not a complex Banach algebra". – user856 Aug 17 '17 at 19:01
• If $A$ is a unital complex algebra, then $\mathbb{C}\cdot 1 \subset Z(A)$ (since $(z\cdot 1)\times a = 1 \times (z\cdot a) = z\cdot a = (z\cdot a)\times 1 = a \times (z\cdot 1)$). But $Z(\mathbb{H}) = \mathbb{R}\cdot 1$. – Daniel Fischer Aug 17 '17 at 19:28

are very important. In general, we define a Banach algebra over a certain field, and the choice of field matters. You can't ignore the specific field, here: $\mathbb{H}$ is a Banach algebra over $\mathbb{R}$, but not over $\mathbb{C}$.