
heilig
holy / sacred
Regarded in a religious sense as divine, consecrated, or worthy of veneration.
Der Ganges ist für Hindus ein heiliger Fluss.
The Ganges is a sacred river for Hindus.
holy / sacred
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holy / sacred
Regarded in a religious sense as divine, consecrated, or worthy of veneration.
Der Ganges ist für Hindus ein heiliger Fluss.
The Ganges is a sacred river for Hindus.

sacred
In a figurative sense, untouchable; something that must not be criticized or changed.
Das Auto ist meinem Vater heilig, da darf niemand sonst mit fahren.
My father's car is sacred to him; no one else is allowed to drive it.

to solemnly promise
Used to strongly affirm a promise and emphasize its seriousness.
Ich verspreche dir hoch und heilig, dass ich es niemandem weitererzähle.
I solemnly promise you that I won't tell anyone.
'Heilig' (holy/sacred) is related to 'das Heil' (well-being, salvation) and the verb 'heilen' (to heal). The basic idea is something whole, unharmed, and worthy of veneration.
Die Pilger suchten an diesem Ort ihr Seelenheil.
The pilgrims sought their spiritual salvation at this place.
In the figurative meaning 'untouchable/sacred', a dative construction is often used. The structure is: [the thing] ist [the person in the dative] heilig.
Die Sonntagsruhe ist meinen Nachbarn heilig.
Sunday rest is sacred to my neighbors.
The expression 'hoch und heilig versprechen' (to solemnly promise) is a fixed idiom. 'Hoch und heilig' functions as a unit here and strongly intensifies the verb 'versprechen' (to promise).
Er hat mir hoch und heilig versprochen, pünktlich zu sein.
He solemnly promised me to be on time.
In a religious context, there are many fixed combinations with 'heilig' (holy/sacred). Well-known examples are 'die Heilige Schrift' (the Holy Scripture; Bible, Quran, etc.), 'ein heiliger Ort' (a holy place), or 'der Heilige Abend' (Christmas Eve, December 24).
Viele Gläubige besuchen die heiligen Stätten in Jerusalem.
Many believers visit the holy sites in Jerusalem.
to swear
In the expression “hoch und heilig versprechen,” heilig reinforces the seriousness of an assurance. Schwören, by contrast, itself means to make a very serious and solemn promise. Use heilig as an intensifier in this fixed expression, while schwören expresses the act of making the promise itself.

Ich verspreche dir hoch und heilig, dass ich niemandem davon erzählen werde.
I solemnly promise you that I won’t tell anyone about it.
Vor Gericht musste der Zeuge schwören, die Wahrheit zu sagen.
The witness had to swear to tell the truth in court.