Metaphysics is the branch of philosophy that deals with being and that tries to answer the “big questions” about life, death, the afterlife, and God. With that definition in mind, we can say that the central conflict of Wole Soyinka's play Death and the King's Horseman is definitely metaphysical.
Elesin is ready to die, for he has a duty that he must fulfill in the afterlife. He is happy to go, but he wants to enjoy a little of life before he does, and he asks the women to dress him in fine clothing. They chant and dance and tell stories that express their beliefs in death and the afterlife.
Elesin also has one more request. He sees a beautiful young woman and asks to marry her. He wants her to conceive his child so that he can leave behind a gift when he goes into the afterworld. At first, the woman refuses but then gives in.
This whole process is supervised by the Praise-Singer, who seems to be a representative of the gods and the afterworld. The Praise-Singer wonders if Elesin is going crazy in his desire to marry this woman, for it is a very bad idea.
The scene then shifts to the English officer's house, and we see Simon Pilkings and his wife, Jane, dressed in traditional costumes that they have no right to wear. They are getting ready to go to a costume ball, but their choice of clothing shows severe disrespect for the culture of the people in whose land they live. Pilkings wants to stop Elesin from killing himself, for he does not understand (and will not even try to understand) the metaphysical beliefs that are driving Elesin. Pilkings has already sent Elesin's son Olunde to England to receive an education and become a doctor, saving him, he thinks, from some ritual that his father insisted he take part in.
Pilkings succeeds in stopping Elesin from dying, although this is a tragedy to the people, for they believe that Elesin is going into the afterlife to save their king. While Elesin is in prison, however, Olunde steps up and dies in his father's place, to Pilkings's horror. Olunde has decided that his traditional beliefs are more important than his education and future career. Elesin, however, is blamed for his refusal to die, as the Praise-Singer recognizes that his desire for his new bride has held him back.
The play's central conflict, then, lies around the difference in the metaphysical beliefs of Elesin and his companions and those of Pilkings. Further, Elesin and Olunde both have to make choices about how they will respond to the demands their beliefs place on them.