Vienna & Zurich
At the turn of the century, there were two schools of psychoanalysis in Europe: the Vienna school (led by Freud) and the Zurich school (ultimately led by Jung). To simplify enormously, Zurich School was directed more towards science and clinical study than Vienna. While a subsidiary character in the film, Eugen Bleuler—the director of the Burghölzi Psychiatric Teaching Clinic in Zurich, where Jung began his training period in 1900—was the leading light of the Zurich school. And at the time, he had a world reputation as high as Freud's. Among many other things, Bleuler coined the term schizophrenia and came up with the idea that patients with acute problems could be helped by being given productive work and responsibilities, and released early. (In A DANGEROUS METHOD we see the effectiveness of these methods on Spielrein.)
To put in context the groundbreaking achievements of Bleuler and Freud, note that only a few decades before, the highly-educated doctors at the Burghölzi—who spoke High German—couldn't communicate at all with their patients, who spoke Low German and a Swiss dialect. At Bleuler's Burghölzi and on Freud's couch, the relationship between doctor and patient was everything.
The scientific approach of the Zurich school is illustrated by the scene in A DANGEROUS METHOD where Jung and Spielrein administer the Word Association Experiment to Jung's wife Emma, utilizing an electro-galvanometer (a device for measuring electric current) to measure her subconscious reactions, and keeping records of her reaction times. (Spielrein did in fact assist him with this test on Emma Jung.) The results of Jung's clinical research with the Word Association Experiment suggested that the "the majority of complexes operative in the association experiments relate to direct or transposed sexuality." This put Jung in a dilemma, as he had independently found through clinical testing a conclusion that Freud had previously published. After some inner debate, Jung chose to acknowledge Freud. In fact, Jung would do much more than that. He soon would be the first to build a bridge between the Vienna and Zurich schools, when he applied the Freudian method on Spielrein. This was not a simple task, as Freud had offered no manual for his interpretation of dreams (other than his master work, Interpretation of Dreams) and only the vaguest outlines of his working methods. It was up to Jung to find his own way into Freudian psychoanalysis, with little but his intuition to guide him.