This could be in a positive or negative way. So someone ignores people telling them to murder or someone ignores someone telling them to stop being rude. Basically how can you describe someone who just follows their gut in everything they do.
ob·sti·nate ˈäbstənət/ adjective stubbornly refusing to change one's opinion or chosen course of action, despite attempts to persuade one to do so.
Taken from Oxford Dictionaries.
What about "maverick"? That's the word that came to my mind, although the following formal definitions don't exactly match your description.
Adjective: Showing independence in thoughts or actions.
Noun: One who does not abide by rules.
Someone who just sticks to their own opinions while ignoring those of others is:
Having an arrogantly high regard for oneself or one’s own opinions:
a pompous, self-opinionated bully
Not a single word, but try:
be your own person/woman/man (Cambridge)
to be in control of your life and not allow other people to tell you what to do:
Nobody tells me how to live my life - I'm my own man.
This sort of independent thinking (good or bad) is shown by someone who is strong-minded.
strong-minded: adjective (Collins)
having strength of mind; firm, resolute, and determined
unreasonably obstinate; obstinately unmoving:
- E.g. a stubborn child.
fixed or set in purpose or opinion; resolute:
- E.g. a stubborn opponent of foreign aid.
obstinately maintained, as a course of action:
- E.g. a stubborn resistance.
difficult to manage or suppress:
- E.g. a stubborn horse; a stubborn pain.
without interest or concern; not caring; apathetic:
- E.g. his indifferent attitude toward the suffering of others.
having no bias, prejudice, or preference; impartial; disinterested.
Virtually all other answers, while certainly apt, connote negatively. Per the OP, please allow me to put a positive spin on this personal characteristic.
independently-minded adjective: self-reliant and seeking autonomy; see, Collins Dictionary
self-reliant adjective: Free from the influence, guidance, or control of others; see, the Free Dictionary
independent adjective: 2. Free from the influence, guidance, or control of another or others; self-reliant: an independent mind. see, the Free Dictionary
strong-minded adjective: 1. Having a determined will. 2. Having a vigorous, independent mind. see, the Free Dictionary
I'd call someone like that an individualist
a person who does things without being concerned about what other people will think
Another option: bull-headed
: not willing to change an opinion, plan, etc. : very stubborn in a foolish or annoying way
Perhaps a more neutral answer is pertinacious.
Like obstinacy and stubbornness, it refers to unshakeable opinions. Unlike those other words, pertinacity can be used in either a positive or negative sense.
1: adhering resolutely to an opinion, purpose, or design
2: stubbornly tenacious
One with a negative connotation:
dog·mat·ic (dôg-măt′ĭk, dŏg-) adj. 1. Relating to, characteristic of, or resulting from dogma. 2. a. Asserting or insisting upon ideas or principles, especially when unproven or unexamined, in an imperious or arrogant manner: "People in recovery groups can be dogmatic, asserting that the group's way is 'the way' or bashing other approaches" (Anne M. Fletcher). b. Characterized by such assertion, often with an unconsidered rejection of criticism: a dogmatic adherence to a single educational model.
Taken from here.
Selfish: having or showing concern only for yourself and not for the needs or feelings of other people Merriam-Webster
The selfish person ignores others and does what they want! This certainly covers the 'negative' scenario of OP's question, but also, the bit about "follow[ing] their gut in everything they do".