Question

Is there an infinite-dimensional, non-commutative complex local algebra $A$ (which is not a field) with the (unique) maximal left-ideal finitely generated as a left ideal? Or as a right ideal?

Answer 1

Yes, the following should do the trick:

Let $D$ be any skew field over $\mathbb{C}$. Then $D[[X]]$ is a local ring with maximal left ideal $(X)$ (see Lam, A first Course in noncommutative Rings, after 19.7) and $D[[X]]$ is clearly infinite-dimensional as $\mathbb C$-algebra.

So it suffices to find $D$. According to Cohn's book quoted in

$\qquad$http://mathoverflow.net/questions/48173/infinite-dimensional-division-algebras,

$D$ can be choosen to be a twisted Laurent series ring over $\mathbb C$ (see Prop. 2.3.5 in google books).