U and V are complex vectors. _n_a will denote the real part and _n_b will denote the complex part.
I'm assuming the law of cosine holds up in complex space:
||U||² + ||V||² - 2*||U||*||V||*cos(angle) = ||U-V||²

||U||² = U_1_a² + U_1_b² + ... + U_n_a² + U_n_b²
||V||² = V_1_a² + V_1_b² + ... + V_n_a² + V_n_b²

||U-V||² = (U_1_a-V_1_a)² + (U_1_b-V_1_b)² + ... + (U_n_a-V_n_a)² + (U_n_b-V_n_b)²
||U-V||² = ||U||² + ||V||² - 2*(U_1_a*V_1_a + U_1_b*V_1_b + ... + U_n_a*V_n_a + U_n_b*V_n_b)

||U||² + ||V||² - 2*||U||*||V||*cos(angle) = ||U-V||²
=
||U||² + ||V||² - 2*(U_1_a*V_1_a + U_1_b*V_1_b + ... + U_n_a*V_n_a + U_n_b*V_n_b)

subtracting ||U||² + ||V||² from both sides gives:
-2*||U||*||V||*cos(angle) = -2*(U_1_a*V_1_a + U_1_b*V_1_b + ... + U_n_a*V_n_a + U_n_b*V_n_b)

dividing both sides by -2 gives:
||U||*||V||*cos(angle) = (U_1_a*V_1_a + U_1_b*V_1_b + ... + U_n_a*V_n_a + U_n_b*V_n_b)

so the complex inner product should be defined by (U_1_a*V_1_a + U_1_b*V_1_b + ... + U_n_a*V_n_a + U_n_b*V_n_b) which doesn't involve any conjugates or the presence of i.