Relevant verses:
2Co 5:1-10 For we know that if the tent that is our earthly home is destroyed, we have a building from God, a house not made with hands, eternal in the heavens. (2) For in this tent we groan, longing to put on our heavenly dwelling, (3) if indeed by putting it on we may not be found naked. (4) For while we are still in this tent, we groan, being burdened–not that we would be unclothed, but that we would be further clothed, so that what is mortal may be swallowed up by life. (5) He who has prepared us for this very thing is God, who has given us the Spirit as a guarantee. (6) So we are always of good courage. We know that while we are at home in the body we are away from the Lord, (7) for we walk by faith, not by sight. (8) Yes, we are of good courage, and we would rather be away from the body and at home with the Lord. (9) So whether we are at home or away, we make it our aim to please him. (10) For we must all appear before the judgment seat of Christ, so that each one may receive what is due for what he has done in the body, whether good or evil.
To be absent from the body, means we die. In the context of 2 Corinthians 5: 6 & 8, to be present with the Lord means that we have been given immortality.
Verses which indicate being absent from the body implies death:
2 Corinthians 5:1-2:
For we know that if the tent that is our earthly home is destroyed, we have a building from God, a house not made with hands, eternal in the heavens. (2) For in this tent we groan, longing to put on our heavenly dwelling
2 Peter 1:13-14:
I think it right, as long as I am in this body, to stir you up by way of reminder, (14) since I know that the putting off of my body will be soon, as our Lord Jesus Christ made clear to me.
2 Timothy 4:6-7:
For I am already being poured out as a drink offering, and the time of my departure has come. (7) I have fought the good fight, I have finished the race, I have kept the faith.
Verses which indicate being present with the Lord implies immortality:
Philippians 1:21-23:
For to me to live is Christ, and to die is gain. (22) If I am to live in the flesh, that means fruitful labor for me. Yet which I shall choose I cannot tell. (23) I am hard pressed between the two. My desire is to depart and be with Christ, for that is far better.
2 Corinthians 4:14:
knowing that he who raised the Lord Jesus will raise us also with Jesus and bring us with you into his presence.
Psalm 17:15:
As for me, I shall behold your face in righteousness; when I awake, I shall be satisfied with your likeness.
2 Timothy 4:7-8:
I have fought the good fight, I have finished the race, I have kept the faith. (8) Henceforth there is laid up for me the crown of righteousness, which the Lord, the righteous judge, will award to me on that Day, and not only to me but also to all who have loved his appearing.
1 John 3:2:
Beloved, we are God’s children now, and what we will be has not yet appeared; but we know that when he appears we shall be like him, because we shall see him as he is.
Revelation 20:6:
Blessed and holy is the one who shares in the first resurrection! Over such the second death has no power, but they will be priests of God and of Christ, and they will reign with him for a thousand years.
To summarise the implications of the above verses:
When we die, we cease to have a conscious existence. Our body reverts to dust and we sleep the sleep of death knowing nothing at all (Eccl. 9:5-6; Psalm 88:10-11). In this state, we wait for the return of Jesus when he will resurrect the dead and give the faithful, immortality. From a believer’s point of view, the death part of this experience, is as though it never happened. From a believer’s point of view, we go straight from our mortal existence, to the resurrection at Jesus’ return — an event yet to occur. So we can say that when we die, we depart and are with Christ in an immortal existence.
We know that we are not given immortality until the resurrection, as we see in the above verses. Other verses which confirm this:
1 Corinthians 15:21-23:
For as by a man came death, by a man has come also the resurrection of the dead. (22) For as in Adam all die, so also in Christ shall all be made alive. (23) But each in his own order: Christ the firstfruits, then at his coming those who belong to Christ.
Isaiah 26:19:
Your dead shall live; their bodies shall rise. You who dwell in the dust, awake and sing for joy! For your dew is a dew of light, and the earth will give birth to the dead.
2 Corinthians 5:10:
For we must all appear before the judgment seat of Christ, so that each one may receive what is due for what he has done in the body, whether good or evil.
What Paul does look forward to is the second coming, when Paul will be both away from his (present suffering, mortal) body and at home with the (returned, triumphant, sovereign) Lord. That is the hope he describes in 2 Cor. 5:8. That is our hope. Not the intermediate state.