Whereas Ashkenazic custom requires standing during Kaddish, the accepted practice among Sepharadim is to remain seated during Kaddish, unless one was standing when Kaddish began, in which case he must remain standing (until he completes his "Yeheh Shemeh Rabba" response).
Hacham Ovadia Yosef ruled that if a Sepharadi wishes to follow the stringent practice of the Ashkenazim and stand for Kaddish, he should not do so, as this appears arrogant. Since the accepted Sephardic custom is to remain seated, and even the greatest Rabbis sit during Kaddish, making a point of standing gives the appearance of thinking that one is better than everyone else, and this is therefore inappropriate. Hacham Ovadia notes the ruling of the Kol Eliyahu that one who decides to stand during the Torah reading – when the commonly accepted custom is to sit – should be reprimanded for this display of arrogance. By the same token, it is inappropriate to break the accepted Sephardic practice by standing for Kaddish.
The Sedeh Hemed (Rav Haim Hizkiya Medini, Hebron, 1834-1904) writes that his practice was to stand for Kaddish. The reason may have been that as a Rabbinic leader, he felt this was an appropriate stringency to accept. Regardless, as a general rule, this is discouraged.
Summary: Sephardic practice is to remain seated during Kaddish, and it is improper for a Sepharadi to make a point of standing.