The bible uses the word "apostle" to mean one who is sent forth with orders. There are two classes of that office. The first are ones who were eyewitnesses to Christ's 3 1/2 year ministry. The second are also messengers of those eyewitness who are called and sent forth to continue the spread of the gospel (emphasis mine).
So, the biblical basis for the continued office of a messenger apostle is not one who adds to or subtracts from the gospel as written during the time of the eyewitness apostles, but as one who faithfully brings the same gospel message forward.
a delegate, messenger, one sent forth with orders Thayers Apostle, Apostleship:
is, lit., "one sent forth" (apo, "from," stello, "to send"). "The word is used of the Lord Jesus to describe His relation to God, Hbr 3:1; see Jhn 17:3. The twelve disciples chosen by the Lord for special training were so called, Luk 6:13; 9:10. Paul, though he had seen the Lord Jesus, 1Cr 9:1; 15:8, had not 'companied with' the Twelve 'all the time' of His earthly ministry, and hence was not eligible for a place among them, according to Peter's description of the necessary qualifications, Act 1:22. Paul was commissioned directly, by the Lord Himself, after His Ascension, to carry the Gospel to the Gentiles. "The word has also a wider reference. In Act 14:4, 14, it is used of Barnabas as well as of Paul; in Rom 16:7 of Andronicus and Junias. In 2Cr 8:23 (RV, margin) two unnamed brethren are called 'apostles of the churches;' in Phl 2:25 (RV, margin) Epaphroditus is referred to as 'your apostle.' It is used in 1Th 2:6 of Paul, Silas and Timothy, to define their relation to Christ." * [* From Notes on Thessalonians, by Hogg and Vine, pp. 59-60.] -Vines-
If you use the word apostle as did Peter in Acts 1:22, then the last apostle is Matthias. He was of the twelve. These were the eyewitness apostles.
Beginning from the baptism of John, unto that same day that he was taken up from us, must one be ordained to be a witness with us of his resurrection.
For Peter, at this time, an apostle was an eyewitness.
Paul, some three years later, was called by God to preach to the gentiles the same gospel message. As well, there were other apostles who spread the same message, but who were not eyewitnesses.
Salute Andronicus and Junia, my kinsmen, and my fellow prisoners, who are of note among the apostles, who also were in Christ before me. Rom 16:7
But other of the apostles saw I none, save James the Lord's brother. Gal 1:19
Which when the apostles, Barnabas and Paul, heard of, they rent their clothes, and ran in among the people, crying out, Acts 14:14
So, the bookends of Christ's specifically called apostles as eyewitness spread from James to John, sons of Zebedee, first and last eyewitness apostles to die. Within this period and continuing to today are faithful apostles called to faithfully spread the exact same gospel to all peoples, even though they are not eyewitnesses.
What about 2 Cor 12:11-12?
I [Paul] am become a fool in glorying; ye have compelled me: for I ought to have been commended of you: for in nothing am I behind the very chiefest apostles, though I be nothing. Truly the signs of an apostle were wrought among you in all patience, in signs, and wonders, and mighty deeds.
The "chiefest" or "pre-eminent" would point to the eyewitness apostles of the 3 1/2 years of Christ's ministry. The twelve and Paul, called of God as an apostle, showed the same signs and wonders among them as the gospel spread from Jerusalem outward. This was God endorsing their gospel. Once the bible was written by the time of John's death, the signs and wonders of the apostolic office was changing from miracles to faithful teaching (a miracle in itself). This is the message of Ephesians 2:20.
And are built upon the foundation of the apostles and prophets, Jesus Christ himself being the chief corner stone;
There are no more eyewitness apostles as also the prophetic office in the Old Testament also ended. What's the contrast?
But there were false prophets also among the people, even as there shall be false teachers among you, who privily shall bring in damnable heresies, even denying the Lord that bought them, and bring upon themselves swift destruction. 2 Peter 2:1
The Old Testament false prophets were those who erroneously spoke in the name of God; likewise, in the New Testament where the apostolic office shifts from speaking as an eyewitness and as from the bible to basically adding to or subtracting from the gospel.
So, what is the biblical basis for the continuation of the apostolic office? If we use that broader definition found in the bible, it becomes faithful teachers of the same Good News.