Magma contains a system [Ste02], [Ste10] for computing with algebraically closed fields, which have the property that they always contain all the roots of any polynomial defined over them. It is of course not possible to construct explicitly the closure of a field, but the system works by automatically constructing larger and larger algebraic extensions of an original base field as needed during a computation, thus giving the illusion of computing in the algebraic closure of the base field.
Such a system was already proposed before (the D5 system [DDD85]), but this has difficulty with the parallelism which occurs when one must compute with several conjugates of a root of a reducible polynomial, leading to situations where a certain expression evaluated at a root is invertible but evaluated at a conjugate of that root is not invertible.
Magma's system has no such problem and one can compute with the field just like any other field in Magma; all standard algorithms which work over generic fields or which use factorization work automatically without having to be adapted to handle the many conjugates of a root.
Especially significant is also the fact that all the Gröbner basis algorithms work well over such fields. One can compute the variety of any zero-dimensional multivariate polynomial ideal over the algebraic closure of its base field. Puiseux expansions of polynomials are also successfully computed using an algebraically closed field.