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?
For motivation, consider the set of all $n \times n$ invertible matrices over a (commutative) field.
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 $\mathbb H$ !
Yes. A lot of interesting number theory is involved. The Brauer group classifies division algebras with a given center, and in class field theory that plays a big role (when the center field is a number field). See for example this question and this.
In addition to number theory, the topic is interesting on its own merits. Over more complicated center fields we no longer get all the division algebras as cyclic algebras. Hopefully some more knowledgable forumite can give you pointers to that theory.
My interest to skewfields (the number theoretic ones in particular) was awakened by the observation that lattices in skewfields yield interesting signal constellations in multi-antenna radio communications. Google for Golden code for the most widely known example.
Damien raised the study of roots of $x^2=-1$ in the ring of quaternions as a question with an unexpected answer. I have written up a shortish related answer. The root cause for the problem is explained in this exposition by Arturo Magidin.