Dot product over complex vectors: Conjugate first or second?

If we consider column vectors in $\mathbb{C}^n$, it's more natural to define the standard inner product by $$ \langle\mathbf{x}\mid\mathbf{y}\rangle = \mathbf{x}^H\mathbf{y} $$ rather than $\mathbf{y}^H\mathbf{x}$ ($H$ denotes the conjugate transpose, notation for the inner product varies among authors), so it's “naturally” antilinear in the first variable and linear in the second.

If one identifies coordinate vectors with rows (writing maps on the right), then the “natural” way becomes the opposite.

It's just a convention; just learn how to translate from one to the other.

In fact I've got a book in which the dot product is considered antilinear in the first argument despite other books having antilinearity in the second one.

However this ambiguity is in no way dangerous, because you may make a composition of operators to have linearity in the argument that you want.

The book in question is Richtmyer, Morton. Difference Methods for Initial-Value Problems