A quick and dirty fork of Wordpress' paginate_links function, modified to output HTML code compliant with the Bootstrap Pagination component.

bootstrap_paginate.php

<?php
/**
 * A quick and dirty fork of Wordpress' paginate_links function, modified to output HTML code
 * compliant with the Bootstrap Pagination component.
 *
 * @global WP_Query   $wp_query
 * @global WP_Rewrite $wp_rewrite
 *
 * @param string|array $args {
 *     Optional. Array or string of arguments for generating paginated links for archives.
 *
 *     @type string $base               Base of the paginated url. Default empty.
 *     @type string $format             Format for the pagination structure. Default empty.
 *     @type int    $total              The total amount of pages. Default is the value WP_Query's
 *                                      `max_num_pages` or 1.
 *     @type int    $current            The current page number. Default is 'paged' query var or 1.
 *     @type bool   $show_all           Whether to show all pages. Default false.
 *     @type int    $end_size           How many numbers on either the start and the end list edges.
 *                                      Default 1.
 *     @type int    $mid_size           How many numbers to either side of the current pages. Default 2.
 *     @type bool   $prev_next          Whether to include the previous and next links in the list. Default true.
 *     @type bool   $prev_text          The previous page text. Default '« Previous'.
 *     @type bool   $next_text          The next page text. Default '« Previous'.
 * 
 *     @type array  $add_args           An array of query args to add. Default false.
 *     @type string $add_fragment       A string to append to each link. Default empty.
 *     @type string $before_page_number A string to appear before the page number. Default empty.
 *     @type string $after_page_number  A string to append after the page number. Default empty.
 * }
 * @return string String of page links or array of page links.
 */
function bootstrap_paginate_links( $args = '' ) {
	global $wp_query, $wp_rewrite;

	// Setting up default values based on the current URL.
	$pagenum_link = html_entity_decode( get_pagenum_link() );
	$url_parts    = explode( '?', $pagenum_link );

	// Get max pages and current page out of the current query, if available.
	$total   = isset( $wp_query->max_num_pages ) ? $wp_query->max_num_pages : 1;
	$current = get_query_var( 'paged' ) ? intval( get_query_var( 'paged' ) ) : 1;

	// Append the format placeholder to the base URL.
	$pagenum_link = trailingslashit( $url_parts[0] ) . '%_%';

	// URL base depends on permalink settings.
	$format  = $wp_rewrite->using_index_permalinks() && ! strpos( $pagenum_link, 'index.php' ) ? 'index.php/' : '';
	$format .= $wp_rewrite->using_permalinks() ? user_trailingslashit( $wp_rewrite->pagination_base . '/%#%', 'paged' ) : '?paged=%#%';

	$defaults = array(
			'base' => $pagenum_link, // http://example.com/all_posts.php%_% : %_% is replaced by format (below)
			'format' => $format, // ?page=%#% : %#% is replaced by the page number
			'total' => $total,
			'current' => $current,
			'show_all' => false,
			'prev_next' => true,
			'prev_text' => __('&laquo; Previous'),
			'next_text' => __('Next &raquo;'),
			'end_size' => 1,
			'mid_size' => 2,
			'type' => 'plain',
			'add_args' => array(), // array of query args to add
			'add_fragment' => '',
			'before_page_number' => '',
			'after_page_number' => ''
	);

	$args = wp_parse_args( $args, $defaults );

	if ( ! is_array( $args['add_args'] ) ) {
			$args['add_args'] = array();
	}

	// Merge additional query vars found in the original URL into 'add_args' array.
	if ( isset( $url_parts[1] ) ) {
			// Find the format argument.
			$format = explode( '?', str_replace( '%_%', $args['format'], $args['base'] ) );
			$format_query = isset( $format[1] ) ? $format[1] : '';
			wp_parse_str( $format_query, $format_args );

			// Find the query args of the requested URL.
			wp_parse_str( $url_parts[1], $url_query_args );

			// Remove the format argument from the array of query arguments, to avoid overwriting custom format.
			foreach ( $format_args as $format_arg => $format_arg_value ) {
					unset( $url_query_args[ $format_arg ] );
			}

			$args['add_args'] = array_merge( $args['add_args'], urlencode_deep( $url_query_args ) );
	}

	// Who knows what else people pass in $args
	$total = (int) $args['total'];
	if ( $total < 2 ) {
			return;
	}
	$current  = (int) $args['current'];
	$end_size = (int) $args['end_size']; // Out of bounds?  Make it the default.
	if ( $end_size < 1 ) {
			$end_size = 1;
	}
	$mid_size = (int) $args['mid_size'];
	if ( $mid_size < 0 ) {
			$mid_size = 2;
	}
	$add_args = $args['add_args'];
	$r = '';
	$page_links = array();
	$dots = false;

	if ( $args['prev_next'] && $current && 1 < $current ) :
			$link = str_replace( '%_%', 2 == $current ? '' : $args['format'], $args['base'] );
			$link = str_replace( '%#%', $current - 1, $link );
			if ( $add_args )
					$link = add_query_arg( $add_args, $link );
			$link .= $args['add_fragment'];

			/**
			 * Filter the paginated links for the given archive pages.
			 *
			 * @since 3.0.0
			 *
			 * @param string $link The paginated link URL.
			 */
			$page_links[] = '<li><a href="' . esc_url( apply_filters( 'paginate_links', $link ) ) . '">' . $args['prev_text'] . '</a></li>';
	endif;
	for ( $n = 1; $n <= $total; $n++ ) :
			if ( $n == $current ) :
					$page_links[] = '<li class="active"><a>' . $args['before_page_number'] . number_format_i18n( $n ) . $args['after_page_number'] . "</a></li>";
					$dots = true;
			else :
					if ( $args['show_all'] || ( $n <= $end_size || ( $current && $n >= $current - $mid_size && $n <= $current + $mid_size ) || $n > $total - $end_size ) ) :
							$link = str_replace( '%_%', 1 == $n ? '' : $args['format'], $args['base'] );
							$link = str_replace( '%#%', $n, $link );
							if ( $add_args )
									$link = add_query_arg( $add_args, $link );
							$link .= $args['add_fragment'];

							/** This filter is documented in wp-includes/general-template.php */
							$page_links[] = "<li><a href='" . esc_url( apply_filters( 'paginate_links', $link ) ) . "'>" . $args['before_page_number'] . number_format_i18n( $n ) . $args['after_page_number'] . "</a></li>";
							$dots = true;
					elseif ( $dots && ! $args['show_all'] ) :
							$page_links[] = '<span class="dots">' . __( '&hellip;' ) . '</span>';
							$dots = false;
					endif;
			endif;
	endfor;
	if ( $args['prev_next'] && $current && ( $current < $total || -1 == $total ) ) :
			$link = str_replace( '%_%', $args['format'], $args['base'] );
			$link = str_replace( '%#%', $current + 1, $link );
			if ( $add_args )
					$link = add_query_arg( $add_args, $link );
			$link .= $args['add_fragment'];

			/** This filter is documented in wp-includes/general-template.php */
			$page_links[] = '<li><a href="' . esc_url( apply_filters( 'paginate_links', $link ) ) . '">' . $args['next_text'] . '</a></li>';
	endif;
	$r .= '<ul class="pagination">' . "\n\t";
	$r .= join( "\n\t", $page_links );
	$r .= "\n" . '</ul>' . "\n";
	return $r;
}

Do you know how this could be modified for Bootstrap 4?
https://v4-alpha.getbootstrap.com/components/pagination/