Complex quadratic forms have a large isotropic subspace ($\dim E=n-\lceil r/2\rceil$)

A quadratic form of rank $r$ on ${\mathbb{C}}^n$ is congruent to $x_1^2+\cdots +x_r^2$. Pair its nonzero coordinates into vectors $e_{2j-1}+i{e}_{2j}$, and include the radical of a bilinear form. Their span is a totally isotropic subspace and has dimension $\lfloor r/2\rfloor +n-r=n-\lceil r/2\rceil \ge \lfloor n/2\rfloor$. The form vanishes on every vector of this subspace, not merely on its basis vectors.

Any $d$ quadratic forms on ${\mathbb{C}}^n$ have a common nonzero zero when $n\ge 2^d$. Choose a totally isotropic subspace for the first form, of dimension at least $\lfloor n/2\rfloor$. Restrict the remaining forms to it and repeat by induction. After $d$ stages at least one dimension remains. This elementary sufficient bound does not assert that $2^d$ is the best possible threshold.