Let a quadratic form be considered compatible with an algebra over a field if they together form a composition algebra. Can an algebra have multiple different quadratic forms that are compatible with the algebra?
Every quadratic form on the complex numbers that is compatible with them is the square of the absolute value.
Proof: The quadratic form evaluated at $1$ must be $1$, and so any root of unity must have quadratic form $1$ because the form must be real. Since the form is continuous, the numbers with form $1$ are the numbers on the unit circle, so the form is the square of the absolute value. This proves that the only continuous quadratic form that is compatible is the absolute value. Since every quadratic form is continuous, the statement above is true.