Does our conscience continue on after we die?

There are people who have had near death experiences who claim that it does.  There is however no proof or compelling evidence that these experiences are anything but random neuron firings experiences by a brain struggling as it slowly dies from lack of oxygen.

Pending quality evidence, we have to assume that the answer is 'no'.

Here is the truth: The Lord Jesus Christ is God. Nobody can be good enough to avoid hell. The truth is that death leads to immediate heaven or hell, depending only on whether the person believed on the Lord Jesus Christ, or not. The truth is that the Lord Jesus Christ loves you and wants to bless your life freely :) Most of all, Jesus wants you with Him forever, and not in hell.
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Believing on the Lord Jesus Christ means being forgiven all sins past and future and means going to heaven. The truth is that every belief except one will lead to eternal torment in the lake of fire for every person. Because nothing pays for our sins except the death and blood of Jesus, the sacrifice of Jesus that is already accomplished by Him . Jesus loves you! The truth is that the Lord Jesus Christ is God, and Jesus died on the cross to pay for all of our sins in full, and then Jesus was buried, and then Jesus resurrected from the dead. Nothing else pays for our sins, not works, not deeds, not religions. So the only way to heaven and to avoid hell, is by believing on Jesus for eternal life (John 6:47), without adding any of your own works (Romans 4:5). Believe in the Lord Jesus Christ to take you to heaven, and you will be in heaven, no matter what, guaranteed. That easy, thanks to Jesus! Tell Jesus that you thank Him that you will be with Him in heaven

No.After a person dies he is no longer conscious of anything at all-Ecclesiastes 9:5.

There's a good novella about this, written by C. S. Lewis (Chronicles of Narnia author).  The novella about what happens to someone who goes to heaven is titled "The Great Divorce" and is an easy read, about 120 pages soaking wet.

The only thing that dies is the flesh body.
The soul, spirit and conscience does not die.
The soul upon death of the flesh body returns to God in the spirit body.
God is not the God of the dead but of the living. Mark 12:27.

Mark 12:27 “He is not the God of the dead, but the God of the living: ye therefore do greatly err.”

Ecclesiastes 12:7 "Then shall the dust return to the earth as it was: and the spirit shall return unto God who gave it.”

The truth is that death leads to heaven or hell, and the only way to heaven is through faith in Jesus Christ.

The Bible is very clear that no one can earn their way to heaven since good works do not pay for sins. See Gal. 2:16, Rom. 3:23-28, 4:4-5, Eph. 2:8-9, Tit. 3:5. The only way to heaven is by believing in Jesus for His free gift of eternal life. Jesus shed His blood on the cross as the full payment for our sins. He did all the work; all He requires of us is that we believe in Him. Once you believe in Jesus Christ to save you, you are secure forever!

Because we have all sinned against a perfectly holy and righteous God, we all deserve eternal separation from God in hell (Rom. 6:23). However, God loves us so much that He sent His only Son Jesus to die on the cross as our Substitute so we don't have to go to hell. Christ died for our sins, was buried, and rose from the grave three days later (1 Cor. 15:3-4). He gives eternal life freely to anyone who simply believes in Him for it. No one can be saved by doing good works. Salvation is by grace alone through faith alone in Christ alone (Eph. 2:8-9).

"Verily, verily, I say unto you, He that believeth on me hath everlasting life" (John 6:47).

Ecclesiastes 9:5,10 says: "For the living know that they will die, but the dead know nothing at all, nor do they have any more reward, because all memory of them is forgotten."; "Whatever your hand finds to do, do with all your might, for there is no work nor planning nor knowledge nor wisdom in the Grave, where you are going."

No.  All the medical evidence of about a century indicates consciousness is generated by a working brain, and when the brain stops working the consciousness goes away.

Dying is simply a cessation of communication, in this case, with this world.

Afer death comes The Judgment.

I think so.