Is there an infinitedimensional, noncommutative complex local algebra $A$ (which is not a field) with the (unique) maximal leftideal finitely generated as a left ideal? Or as a right ideal?
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 infinitedimensional as $\mathbb C$algebra. So it suffices to find $D$. According to Cohn's book quoted in $\qquad$http://mathoverflow.net/questions/48173/infinitedimensionaldivisionalgebras, $D$ can be choosen to be a twisted Laurent series ring over $\mathbb C$ (see Prop. 2.3.5 in google books). 

