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When first starting a character your given full value on the class hit die + your constitution modifier, e.g. a first level monk's HP die is 1d8+con, lets say 2 for this example, which gives him 10 total at first level. Is this also applied when starting a character in a multiclass? Let's say when leveling up my monk decides to take a dip into fighter, is he awarded the max value of 12 (1d10+2) as a starting 1st level fighter?

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No.

You gain hitpoints for your new class as described for levels after 1st (PHB 163)

So you only get the rolled/high average amount, not the full value.

When your character dips into fighter he'll get either

  • result of d10 + 2 HP
    OR
  • 8 HP
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