Does this modification of Gram matrix have a particular name?

Question

If we have $m$ $n$-dimensional vectors $\mathbf{u}_1,...\mathbf{u}_m$, the matrix of their inner products is a Gram matrix. For real vectors $\mathbf{v}$ and $\mathbf{w}$, the inner product is the sum of products of their coordinates. For complex vectors, the inner product is typically the sum of products of conjugated coordinates of the first vector and coordinates of the second vector. If, however, we use the sums of products of coordinates of two complex vectors instead of the inner products, does the matrix of such sums of products for complex vectors have a particular name? I would also appreciate references.