If your space is real, you only have real numbers to play with. That's why the usual inner product on $\mathbb R^n$ is defined as $$\langle x,y\rangle=\sum_{k=1}^n x_ky_k.$$
That's not the case as soon as you have complex numbers at hand. On $\mathbb C^n$, you usually define $$ \langle x,y\rangle=\sum_{k=1}^n x_k\overline{y_k}. $$ The theory of Hilbert spaces and its linear operators is more neat and complete on complex spaces.