Inner products over fields other than $\mathbb{R}$ or $\mathbb{C}$

Question

Do inner products over other fields even make sense? E.g mod 7 or something. Also if I have a complex vector space $V$ it is a also a vector space over the reals but has twice the dimension e.g $\mathbb{C^2}$ has basis{$\begin{pmatrix} 1\\ 0\end{pmatrix}$,$\begin{pmatrix} 0\\ 1\end{pmatrix}$} over $\mathbb{C}$ which is an orthonormal basis? However over $\mathbb{R}$ a basis is{$\begin{pmatrix} 1\\ 0\end{pmatrix}$,$\begin{pmatrix} 0\\ 1\end{pmatrix}$\begin{pmatrix} i\\ 0\end{pmatrix}$,$\begin{pmatrix} 0\\ i\end{pmatrix}. Is this an orthonormal basis ? How are we defining the inner product in each case?