The most common and most likely answer to this crossword clue is the 4 letter word HIGH
I've seen this crossword clue in The Sunday Telegraph
H
I
G
H
More clues leading to the result HIGH
- School broadcast welcome
- Exciting thing a chore? That's not right — very exciting!
- Dear mortal
- Supposedly superior part of Ginger Baker’s kit
- Tango leaves part of leg extended above the normal level
- Nobly inclined to stop exalted lady being heartless
- Emphatically welcomes game after school
- How to win at cards using arbitrary tactics?
- Big increase in pay in this building?
- Showing arms raised arrogantly
Buddy explains!

The straight part
We know that "Off/at altitude" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case off/at altitude = high.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does HIGH mean?
HIGH noun- A high point or position, literally or figuratively; an elevated place; a superior region; a height; the sky; heaven.
- A point of success or achievement; a time when things are at their best.
- A period of euphoria, from excitement or from an intake of drugs.
- A drug that gives such a high.
- A large area of elevated atmospheric pressure; an anticyclone.
- The maximum value attained by some quantity within a specified period.
- The maximum atmospheric temperature recorded at a particular location, especially during one 24-hour period.
- The highest card dealt or drawn.
- To rise.
- Very elevated; extending or being far above a base; tall; lofty.
- Relatively elevated; rising or raised above the average or normal level from which elevation is measured.
- Having a specified elevation or height; tall.
- Elevated in status, esteem, prestige; exalted in rank, station, or character.
- Of great importance and consequence: grave (if negative) or solemn (if positive).
- Consummate; advanced (e.g. in development) to the utmost extent or culmination, or possessing a quality in its supreme degree, at its zenith.
- (in several set phrases) Remote in distance or time.
- (in several set phrases) Very traditionalist and conservative, especially in favoring older ways of doing things; see e.g. high church, High Tory.
- Elevated in mood; marked by great merriment, excitement, etc.
- (of a lifestyle) Luxurious; rich.
- Lofty, often to the point of arrogant, haughty, boastful, proud.
- (with "on" or "about") Keen, enthused.
- (of a body of water) With tall waves.
- Large, great (in amount or quantity, value, force, energy, etc).
- (acoustics) Acute or shrill in pitch, due to being of greater frequency, i.e. produced by more rapid vibrations (wave oscillations).
- Made with some part of the tongue positioned high in the mouth, relatively close to the palate.
- Greater in value than other cards, denominations, suits, etc.
- (of meat, especially venison) Strong-scented; slightly tainted/spoiled; beginning to decompose.
- Intoxicated; under the influence of a mood-altering drug, formerly usually alcohol, but now (from the mid-20th century) usually not alcohol but rather marijuana, cocaine, heroin, etc.
- (of a sailing ship) Near, in its direction of travel, to the (direction of the) wind.
- In or to an elevated position.
- In or at a great value.
- At a pitch of great frequency.
- Thought; intention; determination; purpose.
- To hie; to hasten.