use of complex conjugate of weight vectors in beamforming literature
This comes from the fact that the convolution operator for discrete-time linear systems is typically defined as the inner product between the input signal $x_i(k)$ in your example) and the system's impulse response ($w_i$). The definition of the inner product for complex vectors includes the conjugation:

$$
a \cdot b = \sum a_i b_i^*
$$

For the above definition, note that there doesn't seem to be consistent agreement over which of the arguments to the dot operator gets conjugated, so be careful when reading the literature. For the application of discrete-time FIR filters (which your example is), the typical convention is to conjugate the vector of filter taps, hence the $w_i^*$ term in the sum.

Note that this convention is not universal in the FIR filtering literature; I have seen papers where the convolution sum for a system with complex-valued impulse response does not include the conjugate operator. However, I prefer to use the form that you showed in your question.