Elon Musk and his partner want to name their child X Æ A-12.
Is that name allowed in California, US?
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We could start with what the statutes say (HSC 102425)
(a) The certificate of live birth for a live birth occurring on or after January 1, 2016, shall contain those items necessary to establish the fact of the birth and shall contain only the following information; (1) Full name and sex of the child.
It says nothing about the form of that name. Therefore, any prohibition of the letter Æ (or æ) etc. would have to come from the administrative interpretation of California Department of Public Health, Vital Records. There is a long handbook, which on p. 112 states the rule regarding child names. The rules note that
The form must be completed using the 26 alphabetical characters of the English language with appropriate punctuation, if necessary.
No pictographs (), ideograms (), diacritical marks (è, ñ, ē, ç), or extraneous entries are allowed.
(The pictograms are smiley-face and double-shaft up-arrow). So the short answer is, unless you feel like making a court case out of the matter and you have a lot of money, this name will not be allowed. The rule might be challenged in court as exceeding statutory authority, and might well be deemed to discriminate w.r.t. race and national origin. The rule could be defended on grounds of necessity, if we presume that the department is incapable of recording information other than the 26 letters and "appropriate punctuation" (undefined, presumably only apostrophe). It's not that in principle Unicode doesn't exist, it's that their system may not be capable of dealing with it (numerous problems would arise from the non-unique mapping from visual representation to Unicode number). There does not seem to be a court ruling on the matter.