English Standard Version
but the one who denies [G720] me before men will be denied [G533] before the angels of God.
The two English words for "deny" here are different words in Greek. In fact, the two Greek words are related. G533 is the intense form of G720.
533 aparnéomai (from 575 /apó, "from" which intensifies 720 /arnéomai, "deny") – properly, to deny, looking back to what was originally refused (rejected, forsaken). Note the force of the prefix, 575 /apó ("away from").
720 (arneomai) already means "deny," so 533 (aparnéomai) suggests "strongly reject" (especially the source). That is, utterly refusing to recognize the original source involved. Hence 533 (aparnéomai) can imply "ignore, disown, or repudiate" (Abbott-Smith, so also in Herodotus, Thucydides).
[The high level of personal involvement (interest) motivating 533 (aparnéomai) accounts for why it always in the Greek middle voice.]
We are talking about intense emotion and action. My translation:
but the one who denies me before men will be utterly denied before the angels of God.
Now, let's get on to Matthew 16:24
Berean Study Bible
Then Jesus told His disciples, “If anyone wants to come after Me, he must deny himself and take up his cross and follow Me.
he must deny
Verb - Aorist Imperative Middle - 3rd Person Singular
Strong's 533: From apo and arneomai; to deny utterly, i.e. Disown, abstain.
Reflexive Pronoun - Accusative Masculine 3rd Person Singular
Strong's 1438: Himself, herself, itself.
The verb is already in its apo-intensified form. Add to it another level of intensification by the Middle voice. Add to that another level of intensification by the following word of the Reflexive pronoun. On top of all these, put it in the Imperative mood. Yes, this is extremely serious. My translation:
he must completely and utterly deny his whole being
Would it be more accurate for it to be translated "disown yourselves" or "disassociate with yourselves" or even "disassociate with your own nature?"
"disassociate with your own nature" sounds pretty good to me here.