Positivity of a complex-valued inner product

Question

I was reading through the properties of an inner product, which includes

Positive-definite: $$\langle x,x\rangle >0,\quad x\in V\setminus \{\mathbf {0} \}.$$

This is from Wikipedia. However, given that an inner product can go into any field, including the complex numbers, for example, and there is no total order on the complex numbers, how can we talk about $\langle x,x\rangle >0$? Is the property of positive definiteness actually bipartite, stating both that

1. $\langle x,x\rangle \in \mathbb{R}$, and
2. $\langle x,x\rangle \in \mathbb{R} > 0$?

This makes sense for the complex numbers, but I don't see how it could be generalized to an arbitrary field. Or does positive definiteness just mean nonzero, in the context of general fields?