Why did Eliyahu ask him to make a request?
Malbim: He saw that soon he will depart.
What is "Pi Shenayim b'Ruchacha"?
Radak citing Sanhedrin 47a: It is double 1 what Eliyahu had 2 . Eliyahu revived one Mes, and Elisha revived two - one in Elisha's lifetime, and one after Elisha died. According to the opinion that the one he revived after he died stood on his feet but did not go home (he merely distanced himself from Elisha's grave; this does not count as Techiyas ha'Mesim), he cured Na'aman's Tzara'as. A Metzora is considered dead 3 (Nedarim 64b), for it says "Al Na Sehi ka'Mes" (Bamidbar 12:12).
Radak citing his father: It is two shares of Eliyahu's Ru'ach 4 . This is like "Pi Shenayim b'Chol Asher Yimatzei Lo" (Devarim 21:17), i.e. two shares of his property. The Bechor gets two shares, and [if he has only one brother] his Pashut brother gets the third share.
Malbim #1: He wanted Eliyahu's Ru'ach to rest on him, like it says "Nachah Ru'ach Eliyahu Al Elisha" (verse 15), and Elisha will also retain his previous understanding, so he will have two Ruchos 5 .
Malbim #2: Just like you have double the Ru'ach of other Nevi'im, I should now be at this level. The Mekubalim say that Pinchas is Eliyahu; the Neshamos of Nadav and Avihu were planted in him. He became Kohen in place of them, and received their level. From this was given to him Bris Shalom and life. From this he had double Ru'ach; Elisha requested this.
Malbim: Even though a Navi cannot be Ma'atzil (give a copy of his Ru'ach) more than his own power, Elisha thought that when Eliyahu ascends to Shamayim and removes his garment of physicality, his level will be much greater than now, and if then he will be Ma'atzil on Elisha his light and influence, Elisha's level will be twice Eliyahu's level now, amidst physicality. This is why Eliyahu said that this will be only if he sees him when he is taken, at this level. He can be Ma'atzil onto him only if Elisha is able to see him.
Radak (11): If so, why did Elisha need Eliyahu's scarf to split the Yarden? And even with his scarf, it did not split until he said "where is Hashem, Elokei Eliyahu" (verse 14)!
Sichos Musar (5733, 7) explains that this is because he is unable to give. Even though the laws of Metzora did not apply to Na'aman, perhaps due to his Tzara'as people distanced from him, and he could not give. (PF)
I.e. twice as much as each of Eliyahu's other Talmidim will get.
"B'Ruchacha" implies of your Ru'ach, just like "Pi Shenayim b'Chol Asher Yimatzei Lo" (Devarim 21:17)! Malbim (Bereishis 1:1) explains that the prefix Beis can mean 'Im' (with), and so say Radak (Tehilim 35:16 - "b'Chanfei La'agei Ma'og") and Abarvenel (Shoftim 5:13 - "Yarad Li ba'Giborim.") (PF)
