This may be somewhat related to physics, but I saw in a non-English paper (which I googled "complex $k$-plane" for some constant real $k$) that mentioned a complex $k$-plane.
$k$ in its context was something that was called a "spectral parameter" but namely without detail on some f its property, it's a constant.
In mathematics, when someone says a complex $k$-plane, what does it mean as opposed to a simple complex plane with the real and imaginary axis?
Something pictorial may be great to describe it, I cannot find a clear explanation online.
In physics, the label $k$ (or any other label) for a complex plane is used to indicate that it is being used to model a particular feature of the physical situation at issue, and is used to distinguish that complex plane from other complex planes which are also being used as part of the overall model. One also sees words like "$k$-space." Vague memories of solid state physics come to mind; look at a solid state physics book like Kittel or Ashcroft-Mermin to see instances of how the terminology is used. If your mathematician's instincts find that the physicists' "abuse of language" is too much for you, there's always Reed and Simon's book to explain the same mathematical models of reality in more orthodox language.