What is the hypostatic union? 

In the person of Jesus Christ since His physical birth [incarnation], there are two natures, undiminished deity and true humanity in one person forever. These two natures — human and divine — remain distinct and are inseparably united without mixture or loss of identity, without loss or transfer of attributes. This means that the Lord Jesus Christ is just as much God as God the Father and God the Holy Ghost...and at the same time He is also just as much human as you and I. He is undiminished deity and true humanity in one person.                                                        

It is important to note that the two natures of Christ maintain their complete identity while being joined in personal union forever. The characteristics of His human nature belong to the human part of Him; the characteristics of His divine nature belong to the God part of Him. Each nature has its own attributes that adhere to that nature. In other words, there is no mixture of the two natures. He is never half-God and half-man or half-man and half-God.                                                                                

Sometimes He operates from His human nature only, as in Matthew 4:2:

And when he had fasted 40 days and 40 nights, he was afterward hungry.

John 19:28:

John 14:28, “... for my Father is greater than I."

Sometimes He operates from His divine nature only, saying in John 10:30:

I and my Father are one.

John 5:23, 14:6 and dozens of verses like them describe Jesus' operation using His divine nature only.

Sometimes He operates from both natures together but not mixed when He says such things as in John 14:6 It is impossible to transfer an attribute of one nature to the other nature without destroying that nature.  If God is added to man then man is no longer true humanity!  If man is added to God then God is no longer undiminished deity.  A change of attributes involves a change of essence.  Therefore, there is no change in the essence of deity or the essence of humanity.     

The doctrine of the Hypostatic Union is documented in many passages, including John 1:1‑14, Romans 1:2-5, 9:5, 1 Timothy 3:16, Hebrews 2:14, Philippians 2:5‑11, 1 John 1:1-3, 1 Timothy 6:15. 

The word “hypostatic” comes from Hebrews 1:1-3. The word for image in this passage is the noun hupostasis, which means “substance, essence, standing under or taking a thing upon one­self.”  Jesus Christ took upon Himself true humanity. “Hypostatic” refers to the whole Person of Christ as distinguished from His two natures, divine and human.  The fact that Christ will be true humanity forever indicates that humanity can live forever — but not in the original body, which comes into the world through physical birth; that body is subject to death.  The resurrection body is never subject to any kind of death which is another unique characteristic of Jesus Christ, and the believer will have the same type of body as His - Philippians 3:21.