To assuage my conscience over an unsourced statement in a paper I'm writing:
I am looking for an example of a commutative algebra over the complex numbers having a maximal ideal of codimension >1, or a statement of inexistence.
To assuage my conscience over an unsourced statement in a paper I'm writing: I am looking for an example of a commutative algebra over the complex numbers having a maximal ideal of codimension >1, or a statement of inexistence. 


If the codimension is finite, then there is no such thing. If the codimension can be infinite, then yes, because there are infinite dimensional complex division algebras which are simple, like $\mathbb C(t)$. 


By codimension you just mean as a $\mathbb{C}$vector space? Take the rational function field $\mathbb{C}(t)$. (Note: by the Nullstellensatz, it is not possible to do so with a finitely generated $\mathbb{C}$algebra.) 

