A common point of confusion is that while a complex line has dimension one over C (hence the term "line"), it has dimension two over the real numbers R, and is topologically equivalent to a real plane, not a real line.
In discussions about the properties of a complex line, confusion may stem from the fact that although it is called a "line" because of its one-dimensional nature in the complex setting (C), when considered over the real numbers R, it has two dimensions and is topologically equivalent to a real plane rather than a real line.