“Mother fixation” refers to a person’s excessive dependence on the mother as a result of a mother’s indulgence of her own bonds of affection, resulting in wanting to act like a mother or always relying on mother figures to make decisions.“Oedipal complex” refers to the (unconscious) fear of a son that his father will castrate him because he desires his mother. For Freud, "being a man" depends on a young boy giving up his desire for his mother and, later in life, choosing a differentwoman to love. An “Oedipus complex” generally implies thata personremains inappropriately attached to his mother out of a desire for her and, as a result, afraid ofmale authority figures. Although both terms imply a lack of manliness,“mother-fixation” does not necessarily involve the sexual desire for the mother implied in “Oedipal complex.” In considering these terms as tools of psychological literary analysis, the crucial point would be making this distinction. Applying these terms to Pygmalion would involve turning to the biography of Shaw to determine whether he experienced either of these while growing up, for if he did, we might see traces of these issues in his character, or we might simply look at the character: does Higgens’ behavior suggest he grew up overly-dependent on his mother?That he retains an inappropriate desire for her?Is he fearful of other men?
I tried very hard to locate information about this topic for you, but there is not much to be had on Higgins and the Oedipal complex. I did find one article, from 1958, titled "Shaw's Childhood and Pygmalion," in which the author, Philip Weissman, argues that Shaw's early life is reflected in the character of Henry Higgins. The full article, however, is not available without subscription.
You may be able to create an argument about Shaw filtering his own problems through Higgins by examining the author's early life. Another source reports that, "Lucinda Elisabeth (Gurly) Shaw, his mother, was the daughter of an impoverished landowner. She was 16-years younger than her husband. George Carr was a drunkard - his example prompted his son to become a teetotaller. When he died in 1885, his children and wife did not attend his funeral. Young Shaw and his two sisters were brought up mostly by servants. Shaw's mother eventually left the family home to teach music, singing, in London. When she died in 1913, Shaw confessed to Mrs. Patrick Campbell: "I must write to you about it, because there is no one else who didn't hate her mother, and even who doesn't hate her children."