1. Every once and a while, I'll see something one of my step-daughters does and say to myself, "She is so like her mom!" Then a little bit later, my other step-daughter might do something and I'll think, "She is so like her mom!" The funny thing is, these two daughters are strikingly different from each other!
And so it seems to be with the "deposit" that God has left to us in the Church through the apostles. You hear two different groups of people in the church pointing to quite different aspects of the church today as the modern equivalent of the apostles. As usual, I think they're both right!
2. So the traditional view is this. The apostles were foundational to the church (Eph. 2:20). Who were these apostles? They were, first of all, the twelve, the disciples. These twelve had been with Jesus from the time of John's baptism up through the time of the resurrection (Acts 1:21-22). Then there was another layer of apostles, people like Paul, Barnabas (1 Cor. 9:5-6), Junias, and Andronicus (Rom. 16:7). The risen Jesus had appeared to them and commissioned them to go as witnesses to the resurrection (1 Cor. 9:1).
As the apostles began to die off, they were understood to have left a "deposit" to the next generation (2 Tim. 1:14). This is the way I understand Ephesians 2:20--the apostles and prophets in question were foundational ones. I believe that prophecy remains a gift in the church. But the prophets of Ephesians 2, I believe, were some of the unspoken heroes behind what we now see in Scripture. God used them to guide the earliest church to passages in the Old Testament about Jesus, for example. We may even hear some words in the Gospels from the risen Jesus through them (although that's a sticky wicket I don't want to go into!)
Bottom line, the apostles left us the New Testament as the deposit of their foundational understandings and practices. Thus Scripture is probably the most important heir of the apostles!
3. But the apostles also appointed elders and deacons in the churches. In other words, the apostles left church structures in place so that, after they were gone, there would be authority in the church. Indeed, this is one of the primary ways they functioned in the early church. When the Corinthian church was going up in flames, the apostle came in as authority to settle questions and bring discipline.
It is thus no surprise that bishops and official institutional church structure has traditionally been understood as heir to the apostles. Again, there is certainly some truth to this idea. When the General Superintendent of the Wesleyan Church or the General Board or the District Superintendent or the District Board steps into a thorny situation to try to speak God's authority into it, they are functioning in an apostolic kind of way.
4. But there are a number of voices in the church today who see another function of the apostles, especially the apostle Paul, that sometimes seems gravely missing from the church today. They especially take Ephesians 4:11 as a set of functions for the church for all time and not just for that time. For this group, apostles are outside the box, entrepreneur types who are always on the move and who have a heightened power to their ministry. These apostles have a charisma, an anointing on them that has a power other people immediately see in them.
Paul did not see his ministry as one that stayed still. He did not feel called to minister where ministries were already started (Rom. 15:20). He was always pushing the gospel to places it had never been and bringing in people who had not known it. In Acts 10, God uses Peter to bring the gospel for the first time to Gentiles.
Several corners of the church today are speaking of "apostolic ministry" as something the church today is lacking. Once again, they are seeing a dimension of the apostles and wanting to consolidate the church's recognition of the calling of these types of individuals. They are also heirs of characteristics that we see apostles demonstrating in the New Testament.