Taking 1900, the date of the formulation of Planck's blackbody formula, as the birth of quantum mechanics
Electromagnetism and statistical mechanics.
These were the topics where the cutting-edge research in physics was done at the time. The most brilliant theoretical physicists of the 19th century (James Clerk Maxwell, Hendrik Lorentz, Ludwig Boltzmann, Josiah Willard Gibbs...) were doing research in one of the two or both.
Both fields require a mastery of non-trivial mathematics: vector calculus and quaternions (1) for electromagnetism, combinatorics and probability theory for statistical mechanics.
Also, both fields contain deeply counterintuitive ideas.
In electromagnetism: the (absolutely non trivial!) relation between electricity and magnetism, the discovery that light is nothing else than a wave in the electromagnetic field, the fact that this wave was able to propagate through empty space...
In statistical mechanics: the fact that it was possible to obtain physical laws by applying probability theory to large systems, the connection between the time-reversible microscopic dynamics and the irreversibility in the macroscopic world, the idea itself that matter was composed of atoms and molecules, which was not at all widely accepted in the scientific community of the 19th century.
Quantum mechanics and special relativity were born from research in one of these two fields: think about the blackbody problem (statistical mechanics), the research on the "luminiferous ether" that led to the Michelson-Morley experiment, Einstein's famous paper On the Electrodynamics of Moving Bodies...
(1) Quaternions, an extension of the complex number, were much used in electromagnetism at the time.