Question

What field of science investigates ideas about a possible extraterrestrial origin of life?

Answer 1

The relevant field is astrobiology, the most highly funded branch of science that has the least scientific evidence supporting it.  It encompasses cosmology, chemistry, geology, astrophysics, and biology predominantly and is exploring and attempting to generate knowledge on every aspect of naturalistic investigation of panspermia, habitability of exoplanets, energy sources in the cosmos, potential astro-intelligence, and composition of meteors, comets, asteroids, planets, solar systems and galaxies. The scope of astrobiology must accommodate virtually all of physical reality to be truly comprehensive.

Astrobiology is not funded well enough for the scope of the task, but is driving much of the science universe exploration projects to try to answer one of the fundamental questions of human history, is there other life and how did life come into being. Panspermia, non-directed (biochemical compounds from extraterrestrial sources of any type) and directed (intentional or non-intentional alien transmitted sources) are a prime astrobiological theoretical field and the search for extraterrestrial intelligence,SETI,  is also a prime focus of astrobiology,