For complex inner products: According to Wikipedia its Linearity in the first argument: (https://en.wikipedia.org/wiki/Inner_product_space)
I have seen some place where it is defined conjugate linear in first factor. https://www.youtube.com/watch?v=Kr3X1Pa9N3E&t=505s. Time 15:16 This is one reference only. i have seen other references also.
Am i missing something here or are they defining two different inner products? I really appreciate any help on clarifying this.
It's true that a complex inner product by one of the definitions will not be a complex inner product by the other. But the definition one uses is just a matter of convention: One can freely translate between the two conventions by mapping a complex inner product $(x, y) \mapsto \langle x , y \rangle$ to the complex inner product (with the opposite convention) $(x, y) \mapsto \langle y, x \rangle$.