My question is two part. First, how does the definition of tensors and tensor spaces change when the vectors that the tensors act upon are elements of a complex vector space as apposed to when they are elements of a real vector space? For example the dot product is bilinear when operating on real vectors, but it becomes antilinear in the second argument when applied to complex vectors. Is there a nice generalization of this kind of change when talking about complex vector spaces?
Second, is there any known application of tensors when generalized to complex vector spaces?
My background: I am going to be a sophomore in college next year, majoring in applied mathematics. I have some background in proof based linear algebra.