Probably derived from the phrase to bugger about, also to bugger around, meaning to cause (someone) difficulty or inconvenience, to interfere or meddle with (something), the noun embuggerance, also imbuggerance, denotes something random or unforeseen that complicates a proposed course of action.

However, particularly in Australian English, with reference to the phrase not to care a bugger, also not to give a bugger, meaning not to care at all, the noun imbuggerance, also embuggerance, denotes absolute indifference.

In A Dictionary of Australian Colloquialisms (Sydney University Press in association with Oxford University Press Australia, 1990), Gerald Alfred Wilkes (1927-2020) defined the noun imbuggerance as denoting an intensified form of indifference; G. A. Wilkes quoted the following from The Australian Language: An examination of the English language and English speech as used in Australia, from convict days to the present, with special reference to the growth of indigenous idiom and its use by Australian writers (Sydney: Currawong Publishing, 1966), by Sidney John Baker (1912-1976):

imbuggerance: Indifference strengthened—by allusion to the phrase don’t care a bugger.

In the sense of absolute indifference, the noun imbuggerance, also embuggerance, occurs, for example in the following:

1-: From Vulgar’s the Word, by the Australian novelist, short-story writer, poet and playwright Harold Edward ‘Hal’ Porter (1911-1984), published in The Bulletin (Sydney, New South Wales) of Wednesday 17^th May 1961:

The widow and the deserted wife, the two women and the bottle of gin and 11 o’clock that Sunday morning.
[…]
“Actually, how long’s it been since, Rube?” said Mrs Fender hostessishly peahenning about in sleeveless black with white lapels wider and pointier than a television ephemerid’s, but at rest within, harmless. Women are most harmful when their bodies think they are at rest and they think they are thinking. She was the widow, the Hair Styliste and Beautician, with hair dyed the color of, and a voice like, apricot jam. She was sloshing over-frank gins into tumblers that had once held gherkins or cocktail onions.
“Squash, Rube, or bitter lemon?”
“Matter’f embuggurance [sic], Vi,” forcefully said Rube, deserted wife, muscular false missus of a creature, and zealously conscious of it for years.

2-: From I Am The Way, a letter to the teachers of New South Wales, by an unemployed teacher called Daniel Reidy, published in Education: Journal of the New South Wales Teachers Federation (Sydney, New South Wales) of Monday 13^th September 1982:

You have no friend or saviour anywhere but in me. I am none of your fat and happy and mediocre run-of-the-mill Presidents. I am your natural leader and you know it. I am not going to be forever hanging around here waiting for you to get your act into gear. […]
Whether you elect me or not is to me and my equanimity and serenity a matter of supreme imbuggerance—but reject me or even ignore me at your peril!

3-: From Waffling in White Australia, Chapter 3 of The Happy Isles of Oceania: Paddling the Pacific (New York: G. P. Putnam’s Sons, 1992), by the U.S. novelist and travel writer Paul Theroux (born 1941):

Australia, the gigantic Pacific island in Meganesia, is an underdeveloped country that is bewildered and at times terrified by its own emptiness. People shout as though to keep their spirits up. G’day! How’s your rotten form! Good on yer!
As on every other Oceanic island, most of its people live at its shores and beaches, so its edge is bricked and bungaloid, the rest an insect-haunted wilderness of croaking wind and red desert.
And the whole place is fly-blown. “It seems an unfortunate habit of Australians to speak through their teeth as if they came from the fly country,” a Sydney Judge remarked about fifty years ago, and he went on, “afraid to open their mouths for fear of flies.” People blink at the flies and turn their backs to the desert, showing total native imbuggerance, and say, Go and have a roll! or Who gives a stuff?—in snarly voices. And maybe Australians talk a lot louder because they are so far away from the rest of the world. How else will anyone hear?