If a sesquilinear form is a generalization of a bilinear form, what is a generalization of a sesquilinear form ?
I don't like this reasoning
There’s no need for a generalization of a sesquilinear form. They exist to fix a certain issue with bilinearity in the context of inner products. An inner product needs to have v.v greater than 0 for nonzero vectors. If we assume full bilinearity with complex coefficients, then (iv).(iv) =(i.i)(v.v) = -v.v
This then shows that (iv).(iv) is strictly negative. This is an issue. Sesquilinearity fixes this. There’s then no defect to fix. Mathematicians don’t just event things on what if adventures. There needs to be a reason, and there’s no reason to expand beyond sesquilinearity in this context.
because in my opinion restrict sesquilinear form to correct some bilinear form issues is deprecated.
If by hypothesis a generalization makes sense for a different reason, why would you use this generalization?
However, how these forms can be generalized?
One generalisation would be to replace $i \mapsto -i$ and the identity by some other field automorphisms:
Let $F$ be a field, $V$ an $F$-vector space, and $\alpha,\beta$ automorphisms of $F$. Then we can define a map $\gamma \colon V \times V \to F$ to be $(\alpha,\beta)$(linear? colinear? equilinear? something else?) if $$ \gamma(\lambda u,v) = \alpha(\lambda)\gamma(u,v) , \\ \gamma(u+v,w) = \gamma(u,v) + \gamma(v,w) , \\ \gamma(u,\lambda v) = \beta(\lambda) \gamma(u,v) , \\ \gamma(u,v+w) = \gamma(u,v)+\gamma(u,w) . $$ This is consistent with the vector space axioms since $\alpha,\beta$ are automorphisms.
One could go further, to multi-not-quite-linear maps, by adding more automorphisms and more arguments.
The disadvantage to this approach is that there are no nontrivial automorphisms of $\mathbb{Q}$ or $\mathbb{R}$, and the only well-behaved automorphism of $\mathbb{C}$ is the one that sends $i \mapsto -i$, i.e. complex conjugation. Therefore there's not an awful lot of promise in this extension for the three most commonly used fields.