A.T. Robertson Commentary John 4:47

"When he heard that Jesus was come out of Judaea into Galilee, he went unto him, and besought [him] that he would come down, and heal his son; for he was at the point of death." — John 4:47 (ASV)

When he heard (ακουσας). First aorist active participle of ακουω. The news spread rapidly about Jesus.

Was come (ηκε). Present active indicative of ηκω, one of the perfective presents, retained in indirect discourse. He had heard the people talk about the miracles in Jerusalem and the first one in Cana.

Went and besought (απηλθεν κα ηρωτα). Ingressive aorist indicative (went off at once) and imperfect active (ηρωτα, began to beg and kept it up).

That he would come down (ινα καταβη, ινα and second aorist active subjunctive of καταβαινω, come down at once)

and heal his son (κα ιασητα αυτου τον υιον, ινα construction, sub-final use or object clause, with first aorist middle subjunctive of ιαομα, completely heal).

For he was at the point of death (ημελλεν γαρ αποθνησκειν). Reason (γαρ) for the urgency. Imperfect active of μελλω with present active infinitive old and common verb for what is about to be and it is used with the infinitive present as here, the aorist infinitive (Revelation 13:16), or the future infinitive (Acts 11:28). The idiom is used of the impending death of Jesus (John 11:51; John 12:33; John 18:32).