Let us consider two vectors $u$ and $v$ are in the complex vector space. The inner product of these two vectors is defined by
$u\cdot v=(u_1^*\cdot v_1, u_2^*\cdot v_2,\cdots\cdots, u_n^*\cdot v_n)$
How can this definition be proved? How did we even get to this definition?
As was pointed out in a comment above, $\sqrt{v\cdot v}$ is the length of $v$ in a real vector space (say $\mathbb{R}^n$). Maintaining the same relationship in a complex vector space leads to your definition, as follows. First consider the (one-dimensional) complex vector space $\mathbb{C}$. If $v=a+bi\in\mathbb{C}$, then we want the length of $v$ again to be $\sqrt{v\cdot v}$. One (the only?) reasonable definition of the dot product that gives $$\sqrt{v\cdot v} = \sqrt{a^2+b^2}$$ is $v\cdot v = (a+bi)(a-bi) = v\bar{v}$. It's easy to see that this generalizes to $n$-dimensional complex vector spaces.
Of course, one has to go on to show that this is in fact an inner product. But that's where the intuition came from.