Let $V$ be a vector space over a field $k$. The definition of quadratic form on nLab is a map $q:V\to k$ such that $q(tv)=t^2q(v)$ for all $t\in k,v\in V$, and such that the map $(v,w)\mapsto q(v+w)-q(v)-q(w)$ is a bilinear form. Let's call this a standard quadratic form.
Meanwhile, in appendix of Brian C. Hall's Quantum Theory for Mathematicians, quadratic forms on a complex Hilbert space are defined as a map $q:H\to\mathbb C$ such that $q(\lambda\psi)=|\lambda|^2q(\psi)$ for all $\lambda\in\mathbb C,\psi\in H$, and such that the map $(\psi,\phi)\mapsto\frac12\big(q(\psi+i\phi)-q(\psi)-q(i\phi)\big)-i\frac12\big(q(\psi+\phi)-q(\psi)-q(\phi)\big)$ is a sesquilinear form. Let's call this a hermitian quadratic form.
These do not seem to be equivalent objects: for example, a standard quadratic form over a complex Hilbert space does not give the relation $q(\lambda\psi)=|\lambda|^2q(\psi)$. But the hermitian quadratic form appears to be a reasonable way to adapt the standard version to the complex case by analogy with how the definition of real inner product is adapted to the complex case.
However, I'm wondering if there's a nice point of view I'm missing here. Is there a more general definition of quadratic form that reduces to the above two definitions under certain circumstances? Or maybe a point of view in which it is clear that the hermitian version is the "correct" way to tweak the definition for the complex case?
If you want a slightly more general notion (though not by far the most general there is), let $K$ be a field and $\iota: x\mapsto \overline{x}$ an involutive automorphism (possibly the identity).
Then if $V$ is a $K$-vector space, a hermitian form on $V$ over $(K,\iota)$ is a bi-additive map $$h: V\times V\to K$$ such that $h(\lambda x,y)=\overline{\lambda}h(x,y)$, $h(x,\lambda y)=\lambda h(x,y)$, and $h(y,x)=\overline{h(x,y)}$.
And you can define a "quadratic hermitian form" (though I would not use that terminology myself) basically as something of the form $x\mapsto h(x,x)$.
Now if $\iota$ is the identity, you recover symmetric bilinear forms and quadratic forms, and if $\iota$ is the complex conjugation on $K=\mathbb{C}$, you recover what you described.
There are more general notions over (not necessarily commutative) rings with involution, which will depend on various parameters.