Do people ever study non-commutative fields?
I've heard of a field, and I've heard of a non-commutative (or "not-necessarily commutative) rings. Do people ever study non-commutative fields?
Yes. They are called division rings or skew fields.
As said Robert Israel, they are called skew fields. But the study of skew fields is very different from commutative fields.
For example, if your field is the Quaternions $\mathbb H$, and you consider the polynomial with real coefficients $\rm X^2 + 1$, it has more than 2 roots in $\mathbb H$ !