Rashi: He wanted to dispel him. Amidst humility, he did not want Elisha to see him being taken.

Malbim: Shelemim (people who reached a high level of perfection) want to influence others as much as possible. As long as Elisha was by Eliyahu, the Talmidim could receive from Eliyahu only via Elisha. A bigger luminary is a conduit for light to pass through it to small stars, just like the earth receives light of the sun at night via the moon. Eliyahu wanted that all Bnei ha'Nevi'im will receive his light and Ru'ach, not only Elisha. This is like [Moshe, Eliyahu's] Rebbi said "Mi Yiten Kol Am Hashem Nevi'im Ki Yiten Hashem Es Rucho Aleihem" (Bamidbar 11:29). In Moshe's lifetime, Nevi'im could get an influence of Nevu'ah only via him. After Hashem 'copied' Moshe's Ru'ach onto the 70 elders, [others] received via the Zekenim close to Moshe. They were the conduit for influence to spill to the people. This is like it says (Eruvin 54b), how was the order of learning? Moshe taught Aharon; after, Aharon taught the Zekenim; after, the Zekenim taught Yisrael. When Yehoshua complained that Eldad and Meidad are [prophesizing, i.e.] receiving influence by themselves, he wanted Moshe to stop the influence from them - "Adoni Moshe Kela'em" (Bamidbar 11:29).